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Homotopy theory of higher categoriesCarlos Simpson

To cite this version:Carlos Simpson. Homotopy theory of higher categories. Cambridge University Press, 19, 2011, Newmathematical monographs, 9780521516952. hal-00449826

https://hal.archives-ouvertes.fr/hal-00449826

https://hal.archives-ouvertes.fr

Homotopy Theory

of

Higher Categories

Carlos T. Simpson

CNRS—INSMI

Laboratoire J. A. Dieudonne

Universite de Nice-Sophia Antipolis

[email protected]

Ce papier a beneficie d’une aide de l’Agence Nationale de la Recherche

portant la reference ANR-09-BLAN-0151-02 (HODAG).

This is draft material from a forthcoming book to be published by

Cambridge University Press in the New Mathematical Monographs se-

ries. This publication is in copyright. c©Carlos T. Simpson 2010.

v

Abstract

This is the first draft of a book about higher categories approached

by iterating Segal’s method, as in Tamsamani’s definition of n-nerve

and Pelissier’s thesis. If M is a tractable left proper cartesian model

category, we construct a tractable left proper cartesian model structure

on the category ofM -precategories. The procedure can then be iterated,

leading to model categories of (∞, n)-categories.

Contents

Preface page 1

Acknowledgements 7

PART I HIGHER CATEGORIES 9

1 History and motivation 11

2 Strict n-categories 29

2.1 Godement, Interchange or the Eckmann-Hilton

argument 31

2.2 Strict n-groupoids 33

3 Fundamental elements of n-categories 43

3.1 A globular theory 43

3.2 Identities 46

3.3 Composition, equivalence and truncation 46

3.4 Enriched categories 49

3.5 The (n+ 1)-category of n-categories 50

3.6 Poincare n-groupoids 52

3.7 Interiors 53

3.8 The case n =∞ 54

4 The need for weak composition 56

4.1 Realization functors 57

4.2 n-groupoids with one object 59

4.3 The case of the standard realization 60

4.4 Nonexistence of strict 3-groupoids giving rise to

the 3-type of S2 61

5 Simplicial approaches 69

Contents vii

5.1 Strict simplicial categories 69

5.2 Segal’s delooping machine 71

5.3 Segal categories 74

5.4 Rezk categories 78

5.5 Quasicategories 81

5.6 Going between Segal categories and n-categories 83

5.7 Towards weak ∞-categories 85

6 Operadic approaches 87

6.1 May’s delooping machine 87

6.2 Baez-Dolan’s definition 88

6.3 Batanin’s definition 91

6.4 Trimble’s definition and Cheng’s comparison 95

6.5 Weak units 97

6.6 Other notions 100

7 Weak enrichment over a cartesian model category:

an introduction 103

7.1 Simplicial objects in M 103

7.2 Diagrams over ∆X 104

7.3 Hypotheses on M 105

7.4 Precategories 106

7.5 Unitality 107

7.6 Rectification of ∆X -diagrams 109

7.7 Enforcing the Segal condition 110

7.8 Products, intervals and the model structure 112

PART II CATEGORICAL PRELIMINARIES 115

8 Some category theory 117

8.1 Locally presentable categories 119

8.2 Monadic projection 123

8.3 Miscellany about limits and colimits 128

8.4 Diagram categories 129


8.6 Internal Hom 138

8.7 Cell complexes 139

8.8 The small object argument 155

8.9 Injective cofibrations in diagram categories 157

9 Model categories 163

9.1 Quillen model categories 163

viii Contents

9.2 Cofibrantly generated model categories 166

9.3 Combinatorial and tractable model categories 167

9.4 Homotopy liftings and extensions 168

9.5 Left properness 171

9.6 Quillen adjunctions 175

9.7 The Kan-Quillen model category of simplicial sets 175

9.8 Model structures on diagram categories 176

9.9 Pseudo-generating sets 179

10 Cartesian model categories 192


10.2 The enriched category associated to a cartesian

model category 197

11 Direct left Bousfield localization 198

11.1 Projection to a subcategory of local objects 198

11.2 Weak monadic projection 205

11.3 New weak equivalences 210

11.4 Invariance properties 213

11.5 New fibrations 218

11.6 Pushouts by new trivial cofibrations 220

11.7 The model category structure 221

11.8 Transfer along a left Quillen functor 224

PART III GENERATORS AND RELATIONS 227

12 Precategories 229

12.1 Enriched precategories with a fixed set of objects 230

12.2 The Segal conditions 231

12.3 Varying the set of objects 232

12.4 The category of precategories 234

12.5 Basic examples 235

12.6 Limits, colimits and local presentability 237

12.7 Interpretations as presheaf categories 243

13 Algebraic theories in model categories 253

13.1 Diagrams over the categories ǫ(n) 254

13.2 Imposing the product condition 259

13.3 Algebraic diagram theories 266

13.4 Unitality 268

13.5 Unital algebraic diagram theories 274

13.6 The generation operation 275

Contents ix

13.7 Reedy structures 276

14 Weak equivalences 277

14.1 The model structures on PC(X,M ) 278

14.2 Unitalization adjunctions 281

14.3 The Reedy structure 283

14.4 Some remarks 289

14.5 Global weak equivalences 291

14.6 Categories enriched over ho(M ) 294

14.7 Change of enrichment category 296

15 Cofibrations 300

15.1 Skeleta and coskeleta 300

15.2 Some natural precategories 305

15.3 Projective cofibrations 308

15.4 Injective cofibrations 310

15.5 A pushout expression for the skeleta 312

15.6 Reedy cofibrations 313

15.7 Relationship between the classes of cofibrations 326

16 Calculus of generators and relations 329

16.1 The Υ precategories 329

16.2 Some trivial cofibrations 331

16.3 Pushout by isotrivial cofibrations 335

16.4 An elementary generation step Gen 343

16.5 Fixing the fibrant condition locally 347

16.6 Combining generation steps 347

16.7 Functoriality of the generation process 348

16.8 Example: generators and relations for 1-categories 350

17 Generators and relations for Segal categories 353


17.2 The Poincare-Segal groupoid 355

17.3 The calculus 357

17.4 Computing the loop space 370

17.5 Example: π3(S2) 378

PART IV THE MODEL STRUCTURE 383

18 Sequentially free precategories 385

18.1 Imposing the Segal condition on Υ 385

18.2 Sequentially free precategories in general 386

x Contents

19 Products 396

19.1 Products of sequentially free precategories 396

19.2 Products of general precategories 406

19.3 The role of unitality, degeneracies and higher

coherences 413

19.4 Why we can’t truncate ∆ 416

20 Intervals 418

20.1 Contractible objects and intervals in M 419

20.2 Intervals for M -enriched precategories 421

20.3 The versality property 427

20.4 Contractibility of intervals for K -precategories 429

20.5 Construction of a left Quillen functor K →M 431

20.6 Contractibility in general 432

20.7 Pushout of trivial cofibrations 434

20.8 A versality property 439

21 The model category of M -enriched precategories 441

21.1 A standard factorization 441

21.2 The model structures 442

21.3 The cartesian property 445

21.4 Properties of fibrant objects 446

21.5 The model category of strict M -enriched categories 447

22 Iterated higher categories 448

22.1 Initialization 449

22.2 Notations 449

22.3 The case of n-nerves 450

22.4 Truncation and equivalences 452

22.5 The (n+ 1)-category nCAT 454

References 456

Preface

The theory of n-categories is currently under active consideration by a

number of different research groups around the world. The history of

the subject goes back a long way, on separate but interrelated tracks

in algebraic topology, algebraic geometry, and category theory. For a

long time, the crucial definition of weakly associative higher category

remained elusive, but now on the contrary we have a plethora of different

possibilities available. One of the next major problems in the subject will

be to achieve a global comparison between these different approaches.

Some work is starting to come out in this direction, but in the current

state of the theory the various different approaches remain distinct. After

the comparison is achieved, they will be seen as representing different

facets of the theory, so it is important to continue working in all of these

different directions.

The purpose of the present book is to concentrate on one of the meth-

ods of defining and working with higher categories, very closely based

on the work of Graeme Segal in algebraic topology many years earlier.

The notion of “Segal category”, which is a kind of category weakly en-

riched over simplicial sets, was considered by Vogt and Dwyer, Kan and

Smith. The application of this method to n-categories was introduced

by Zouhair Tamsamani. And then put into a strictly iterative form, with

a general model category as input, by Regis Pelissier following a sugges-

tion of Andre Hirschowitz. Our treatment will integrate important ideas

contributed by Julie Bergner, Clark Barwick, Jacob Lurie and others.

The guiding principle is to use the category of simplices ∆ as the

basis for all the higher coherency conditions which come in when we

This is draft material from a forthcoming book to be published by Cambridge Uni-versity Press in the New Mathematical Monographs series. This publication is incopyright. c©Carlos T. Simpson 2010.

2 Preface

allow weak associativity. The objects of ∆ are nonempty finite ordinals

[0] = υ0

[1] = υ0, υ1

[2] = υ0, υ1, υ2

. . .

whereas the morphisms are nondecreasing maps between them. Kan

had already introduced this category into algebraic topology, consid-

ering simplicial sets which are functors ∆o → Set. These model the

homotopy types of CW-complexes.

One of the big problems in algebraic topology in the 1960’s was to

define notions of delooping machines. Segal’s way was to consider sim-

plicial spaces, or functors A : ∆o → Top, such that the first space is

just a point A0 = A([0]) = ∗. In ∆ there are three nonconstant maps

f01, f12, f02 : [1]→ [2]

where fij denotes the map sending υ0 to υi and υ1 to υj . In a simplicial

space A which is a contravariant functor on ∆, we get three maps

f∗01, f

∗12, f

∗02 : A2 → A1.

Organize the first two as a map into a product, giving a diagram of the

form

A2σ2→ A1 ×A1

A1

f∗02

↓

.

If we require the second Segal map σ2 := (f∗01, f

∗12) to be an isomorphism

between A2 and A1×A1, then f∗02 gives a product on the space A1. The

basic idea of Segal’s delooping machine is that if we only require σ2 to be

a weak homotopy equivalence of spaces, then f∗02 gives what should be

considered as a “product defined up to homotopy”. One salient aspect

of this approach is that no map A1 ×A1 → A1 is specified, and indeed

if the spaces involved have bad properties there might exist no section

of σ2 at all.

The term “delooping machine” refers to any of several kinds of further

mathematical structure on the loop space ΩX , enhancing the basic com-

position of loops up to homotopy, which should allow one to reconstruct

a space X up to homotopy. In tandem with Segal’s machine in which

Preface 3

ΩX = A1, the notion of operad introduced by Peter May underlies the

best known and most studied family of delooping machines. There were

also other techniques such as PROP’s which are starting to receive re-

newed interest. The various kinds of delooping machines are sources for

the various different approaches to higher categories, after a multiplying

effect whereby each delooping technique leads to several different defini-

tions of higher categories. Our technical work in later parts of the book

will concentrate on the particular direction of iterating Segal’s approach

while maintaining a discrete set of objects, but we will survey some of

the many other approaches in later chapters of Part I.

The relationship between categories and simplicial objects was noticed

early on with the nerve construction. Given a category C its nerve is the

simplicial set NC : ∆o → Set such that (NC)m is the set of composable

sequences of m arrows

x0g1→ x1

g2→ · · ·xm−1

gm→ xm

in C. The operations of functoriality for maps [m] → [p] are obtained

using the composition law and the identities of C. The first piece is just

the set (NC)0 = Ob(C) of objects of C, and the nerve satisfies a relative

version of the Segal condition:

σm : (NC)m∼=−→ (NC)1 ×(NC)0 (NC)1 ×(NC)0 · · · ×(NC)0 (NC)1.

Conversely, any simplicial set ∆o → Set satisfying these conditions

comes from a unique category and these constructions are inverses. In

other words, categories may be considered as simplicial sets satisfying

the Segal conditions.

In comparing this with Segal’s situation, recall that he required A0 =

∗, which is like looking at a category with a single object.

An obvious way of putting all of these things together is to consider

simplicial spaces A : ∆o → Top such that A0 is a discrete set—to be

thought of as the “set of objects”—but considered as a space, and such

that the Segal maps

σm : Am → A1 ×A0 A1 ×A0 · · · ×A0 A1

are weak homotopy equivalences for all m ≥ 2. Functors of this kind are

Segal categories. We use the same terminology when Top is replaced

by the category of simplicial sets K := Set∆o

= Func(∆o,Set). The

possibility of making this generalization was clearly evident at the time

of Segal’s papers [185] [187], but was made explicit only later by Vogt

[184] and Dwyer, Kan and Smith [92].

4 Preface

Segal categories provide a good way of considering categories enriched

over spaces. However, a more elementary approach is available, by look-

ing at categories strictly enriched over spaces, i.e. simplicial categories.

A simplicial category could be viewed as a Segal category where the

Segal maps σm are isomorphisms. More classically it can be considered

as a category enriched in Top or K , using the definitions of enriched

category theory. In a simplicial category, the product operation is well-

defined and strictly associative.

Dwyer, Kan and Smith showed that we don’t lose any generality at this

level by requiring strict associativity: every Segal category is equivalent

to a simplicial category [92]. Unfortunately, we cannot just iterate the

construction by continuing to look at categories strictly enriched over the

category of simplicial categories and so forth. Such an iteration leads to

higher categories with strict associativity and strict units in the middle

levels. One way of seeing why this isn’t good enough1 is to note that the

Bergner model structure on strict simplicial categories, is not cartesian:

products of cofibrant objects are no longer cofibrant. This suggests the

need for a different construction which preserves the cartesian condition,

and Segal’s method works.

The iteration then says: a Segal (n+1)-category is a functor from ∆o

to the category of Segal n-categories, whose first element is a discrete set,

and such that the Segal maps are equivalences. The notion of equivalence

needs to be defined in the inductive process [206]. This iterative point

of view towards higher categories is the topic of our book.

We emphasize an algebraic approach within the world of homotopy

theory, using Quillen’s homotopical algebra [175], but also paying par-

ticular attention to the process of creating a higher category from gener-

ators and relations. For me this goes back to Massey’s book [163] which

was one of my first references both for for algebraic topology, and for

the notion of a group presented by generators and relations.

One of the main inspirations for the recent interest in higher categories

came from Grothendieck’s manuscript Pursuing stacks. He set out a wide

vision of the possible developments and applications of the theory of n-

categories going up to n = ω. Many of his remarks continue to provide

important research directions, and many others remain untouched.

The other main source of interest stems from the Baez-Dolan conjec-

tures. These extend, to higher categories in all degrees, the relationships

1 Paoli has shown [170] that n-groupoids can be semistrictified in any singledegree, however one cannot get strictness in many different degrees as will beseen in Chapter 4.

Preface 5

explored by many researchers between various categorical structures and

phenomena of knot invariants and quantum field theory. Hopkins and

Lurie have recently proven a major part of these conjectures. These mo-

tivations incite us to search for a good understanding of the algebra of

higher categories, and I hope that the present book can contribute in a

small way.

The mathematical discussion of the contents of the main part of the

book will be continued in more detail in Chapter 7 at the end of Part

I. The intervening chapters of Part I serve to introduce the problem

by giving some motivation for why higher categories are needed, by

explaining why strict n-categories aren’t enough, and by considering

some of the many other approaches which are currently being developed.

In Part II we collect our main tools from the theory of categories,

including locally presentable categories and closed model categories. A

small number of these items, such as the discussion of enriched cate-

gories, could be useful for reading Part I. The last chapter of Part II

concerns “direct left Bousfield localization”, which is a special case of

left Bousfield localization in which the model structure can be described

more explicitly. This Chapter 11, together with the general discussion

of cell complexes and the small object argument in Chapters 8 and 9,

are intended to provide some “black boxes” which can then be used in

the rest of the book without having to go into details of cardinality ar-

guments and the like. It is hoped that this will make a good part of the

book accessible to readers wishing to avoid too many technicalities of

the theory of model categories, although some familiarity is obviously

necessary since our main goal is to construct a cartesian model structure.

In Part III starts the main work of looking at weakly M -enriched

(pre)categories. This part is entitled “Generators and relations” because

the process of starting with an M -precategory and passing to the as-

sociated weakly M -enriched category by enforcing the Segal condition,

should be viewed as a higher or weakly enriched analogue of the classi-

cal process of describing an algebraic object by generators and relations.

We develop several aspects of this point of view, including a detailed

discussion of the example of categories weakly enriched over the model

category K of simplicial sets. We see in this case how to follow along the

calculus of generators and relations, taking as example the calculation

of the loop space of S2.

Part IV contains the construction of the cartesian model category,

6 Preface

after the two steps treating specific elements of our categorical situation:

products, including the proof of the cartesian condition, and intervals.

Part V, not yet present in the current version, will discuss various

directions going towards basic techniques in the theory of higher cate-

gories, using the formalism developed in Parts II-IV.

The first few chapters of Part V should contain discussions of inversion

of morphisms, limits and colimits, and adjunctions, based to a great

extent on my preprint [194].

For the case of (∞, 1)-categories, these topics are treated in Lurie’s

recent book [153] about the analogue of Grothendieck’s theory of topoi,

using quasicategories.

Other topics from higher category theory which will only be discussed

very briefly are the theory of higher stacks, and the Poincare n-groupoid

and van Kampen theorems. For the theory of higher stacks the reader

can consult [117] which starts from the model categories constructed

here.

For a discussion of the Poincare n-groupoid, the reader can consult

Tamsamani’s original paper [206] as well as Paoli’s discussion of this

topic in the context of special Catn-groups [170].

We will spend a chapter looking at the Breen-Baez-Dolan stabiliza-

tion hypothesis, following the preprint [196]. This is one of the first parts

of the famous Baez-Dolan conjectures. These conjectures have strongly

motivated the development of higher category theory. Hopkins and Lurie

have recently proven important pieces of the main conjectures. The sta-

bilization conjecture is a preliminary statement about the behavior of the

notion of k-connected n-category, understandable with the basic tech-

niques we have developed here. We hope that this will serve as an intro-

duction to an exciting current area of research.

Acknowledgements

I would first like to thank Zouhair Tamasamani, whose original work on

this question led to all the rest. His techniques for gaining access to a

theory of n-categories using Segal’s delooping machine, set out the basic

contours of the theory, and continue to inform and guide our understand-

ing. I would like to thank Andre Hirschowitz for much encouragement

and many interesting conversations in the course of our work on de-

scent for n-stacks, n-stacks of complexes, and higher Brill-Noether. And

to thank Andre’s thesis student Regis Pelissier who took the argument

to a next stage of abstraction, braving the multiple difficulties not the

least of which were the cloudy reasoning and several important errors

in one of my preprints. We are following quite closely the main idea

of Pelissier’s thesis, which is to iterate a construction whereby a good

model category M serves as input, and we try to get out a model cate-

gory of M -enriched precategories. Clark Barwick then added a further

crucial insight, which was that the argument could be broken down into

pieces, starting with a fairly classical left Bousfield localization. Here

again, Clark’s idea serves as groundwork for our approach. Jacob Lurie

continued with many contributions, on different levels most of which

are beyond our immediate grasp; but still including some quite under-

standable innovations such as the idea of introducing the category ∆X

of finite ordered sets decorated with elements of the set X “of objects”.

This leads to a significant lightening of the hypotheses needed of M .

His approach to cardinality questions in the small object argument is

groundbreaking, and we give here an alternate treatment which is cer-

tainly less streamlined but might help the reader to situate what is going

on. These items are of course subordinated to the use of Smith’s recogni-

tion principle and Dugger’s notion of combinatorial model category, on

which our constructions are based. Julie Bergner has gained important


8 Acknowledgements

information about a whole range of model structures starting with her

consolidation of the Dwyer-Kan structure on the category of simplicial

categories. Her characterization of fibrant objects in the model struc-

tures for Segal categories, carries over easily to our case and provides

the basis for important parts of the statements of our main results.

Bertrand Toen was largely responsible for teaching me about model

categories. His philosophy that they are a good down-to-earth yet pow-

erful approach to higher categorical questions, is suffused throughout

this work. I would like to thank Joseph Tapia and Constantin Teleman

for their encouragement in this direction too, and Georges Maltsiniotis

and Alain Bruguieres who were able to explain the higher categorical

meaning of the Eckmann-Hilton argument in an understandable way. I

would similarly like to thank Clemens Berger, Ronnie Brown, Eugenia

Cheng, Denis-Charles Cisinski, Delphine Dupont, Joachim Kock, Peter

May, and Simona Paoli, for many interesting and informative conver-

sations about various aspects of this subject; and to thank my current

doctoral students for continuing discussions in directions extending the

present work, which motivate the completion of this project. I would also

like to thank my co-workers on related topics, things which if they are

not directly present here, have still contributed a lot to the motivation

for the study of higher categories.

I would specially like to thank Diana Gillooly of Cambridge University

Press, and Burt Totaro, for setting this project in motion.

Paul Taylor’s diagram package is used for the commutative diagrams

and even for the arrows in the text.

For the title, we have chosen something almost the same as the title

of a special semester in Barcelona a while back; we apologize for this

overlap.

This research is partially supported by the Agence Nationale de la

Recherche, grant ANR-09-BLAN-0151-02 (HODAG). I would like to

thank the Institut de Mathematiques de Jussieu for their hospitality

during the completion of this work.

PART I

HIGHER CATEGORIES

1

History and motivation

The most basic motivation for introducing higher categories is the ob-

servation that CatU, the category of U-small categories, naturally has

a structure of 2-category: the objects are categories, the morphisms

are functors, and the 2-morphisms are natural transformations between

functors. If we denote this 2-category by Cat2cat then its truncation

τ≤1Cat2cat to a 1-category would have, as morphisms, the equivalence

classes of functors up to natural equivalence. While it is often neces-

sary to consider two naturally equivalent functors as being “the same”,

identifying them formally leads to a loss of information.

Topologists are confronted with a similar situation when looking at

the category of spaces. In homotopy theory one thinks of two homo-

topic maps between spaces as being “the same”; however the homotopy

category ho(Top) obtained after dividing by this equivalence relation,

doesn’t retain enough information. This loss of information is illustrated

by the question of diagrams. Suppose Ψ is a small category. A dia-

gram of spaces is a functor T : Ψ → Top, that is a space T (x) for

each object x ∈ Ψ and a map T (a) : T (x) → T (y) for each arrow

a ∈ Ψ(x, y), satisfying strict compatibility with identities and composi-

tions. The category of diagrams Func(Ψ,Top) has a natural subclass

of morphisms: a morphism f : S → T of diagrams is a levelwise weak

equivalence if each f(x) : S(x) → T (x) is a weak equivalence. Letting

W = WFunc(Ψ,Top) denote this subclass, the homotopy category of

diagrams ho(Func(Ψ,Top)) is defined to be the Gabriel-Zisman local-

ization W −1Func(Ψ,Top). There is a natural functor

ho(Func(Ψ,Top)) → Func(Ψ, ho(Top)),

which is not in general an equivalence of categories. In other words

ho(Top) doesn’t retain enough information to recover ho(Func(Ψ,Top)).


12 History and motivation

Thus, we need to consider some kind of extra structure beyond just the

homotopy category.

This phenomenon occurs in a number of different places. Starting in

the 1950’s and 1960’s, the notion of derived category, an abelianized

version of ho( ), became crucial to a number of areas in modern ho-

mological algebra and particularly for algebraic geometry. The notion of

localization of a category seems to have been proposed in this context by

Serre, and appears in Grothendieck’s Tohoku paper [106]. A systematic

treatment is the subject of Gabriel-Zisman’s book [100].

As the example of diagrams illustrates, in many derived-categorical

situations one must first make some intermediate constructions on un-

derlying categorical data, then pass to the derived category. A funda-

mental example of this kind of reasoning was Deligne’s approach to the

Hodge theory of simplicial schemes using the notion of “mixed Hodge

complex” [80].

In the nonabelian or homotopical-algebra case, Quillen’s notion of

closed model category formulates a good collection of requirements that

can be made on the intermediate categorical data. Quillen in [175] asked

for a general structure which would encapsulate all of the higher homo-

topical data. In one way of looking at it, the answer lies in the notion of

higher category. Quillen had already provided this answer with his defini-

tion of “simplicial model category”, wherein the simplicial subcategory

of cofibrant and fibrant objects provides a homotopy invariant higher

categorical structure. As later became clear with the work of Dwyer

and Kan, this simplicial category contains exactly the right information.

The notion of Quillen model category is still one of the best ways of ap-

proaching the problem of calculation with homotopical objects, so much

so that we adopt it as a basic language for dealing with notions of higher

categories.

Bondal and Kapranov introduced the idea of enhanced derived cat-

egories [135] [43], whereby the usual derived category, which is the

Gabriel-Zisman localization of the category of complexes, is replaced by

a differential graded (dg) category containing the required higher homo-

topy information. The notion of dg-category actually appears near the

end of Gabriel-Zisman’s book [100] (where it is compared with the no-

tion of 2-category), and it was one of the motivations for Kelly’s theory

of enriched categories [139]. The notion of dg-category, now further de-

velopped by Keller [138], Tabuada [203], Stanculescu [199], Batanin [22],

Moriya [169] and others, is one possible answer to the search for higher

categorical structure in the k-linear case, pretty much analogous to the

History and motivation 13

notion of strict simplicial category. The corresponding weak notion is

that of A∞-category used for example by Fukaya [98] and Kontsevich

[144]. This definition is based on Stasheff’s notion of A∞-algebra [200],

which is an example of the passage from delooping machinery to higher

categorical theories.

In the far future one could imagine starting directly with a notion of

higher category and bypassing the model-category step entirely, but for

now this raises difficult questions of bootstrapping. Lurie has taken this

kind of program a long way in [153] [154], using the notion of quasicate-

gory as his basic higher-categorical object. But even there, the underly-

ing model category theory remains important. The reader is invited to

reflect on this interesting problem.

The original example of the 2-category of categories, suggests us-

ing 2-categories and their eventual iterative generalizations, as higher

categorical structures. This point of view occured as early as Gabriel-

Zisman’s book, where they introduce a 2-category enhancing the struc-

ture of ho(Top) as well as its analogue for the category of complexes,

and proceed to use it to treat questions about homotopy groups.

Benabou’s monograph [28] introduced the notion of weak 2-category,

as well as various notions of weak functor. These are also related to

Grothendieck’s notion of fibered category in that a fibered category may

be viewed as some kind of weak functor from the base category to the

2-category of categories.

Starting with Benabou’s book, it has been clear that there would

be two types of generalization from 2-categories to n-categories. The

strict n-categories are defined recurrently as categories enriched over the

category of strict n − 1-categories. By the Eckmann-Hilton argument,

these don’t contain enough objects, as we shall discuss in Chapter 4.

For this reason, these are not our main objects of study and we will

use the terminology strict n-category. The relative ease of defining strict

n-categories nonetheless makes them attractive for learning some of the

basic outlines of the theory, the starting point of Chapter 2.

The other generalization would be to consider weak n-categories also

called lax n-categories, and which we usually call just “n-categories”, in

which the composition would be associative only up to a natural equiv-

alence, and similarly for all other operations. The requirement that all

equalities between sequences of operations be replaced by natural equiv-

alences at one level higher, leads to a combinatorial explosion because

the natural equivalences themselves are to be considered as operations.

For this reason, the theory of weak 3-categories developped by Gor-


don, Power, Street [104] following the path set out by Benabou for 2-

categories in [28], was already very complicated; for n = 4 it became next

to impossible (see however [211]) and development of this line stopped

there.

In fact, the problem of defining and studying the higher operations

which are needed in a weakly associative category, had been considered

rather early on by the topologists who noticed that the notion of “H-

space”, that is to say a space with an operation which provides a group

object in the homotopy category, was insufficient to capture the data

contained in a loop space. One needs to specify, for example, a homotopy

of associativity between (x, y, z) 7→ x(yz) and (x, y, z) 7→ (xy)z. This

“associator” should itself be subject to some kind of higher associativity

laws, called coherence relations, involving composition of four or more

elements.

One of the first discussions of the resulting higher coherence struc-

tures was Stasheff’s notion of A∞-algebra [200]. This was placed in the

realm of differential graded algebra, but not long thereafter the notion

of “delooping machine” came out, including MacLane’s notion of PROP,

then May’s operadic and Segal’s simplicial delooping machines.

In Segal’s case, the higher coherence relations come about by requiring

not only that σ2 be a weak equivalence, but that all of the “Segal maps”

σm : Am → A1 × . . .×A1

given by σm = (f∗01, f

∗12, . . . , f

∗m−1,m) should be weak homotopy equiva-

lences. This was iterated by Dunn [85]. In the operadic viewpoint, the

coherence relations come from contractibility of the spaces of n-ary op-

erations.

By the late 1960’s and early 1970’s, the topologists had their delooping

machines well in hand. A main theme of the present work is that these

delooping machines can generally lead to definitions of higher categories,

but that doesn’t seem to have been done explicitly at the time. A related

notion also appeared in the book of Boardman and Vogt [42], that of re-

stricted Kan complex. These objects are now known as “quasicategories”

thanks to Joyal’s work [126]. At that time, in algebraic geometry, an

elaborate theory of derived categories was being developed, but it relied

only on 1-categories which were the τ≤1 of the relevant higher categories.

This difficulty was worked around at all places, by techniques of work-

ing with explicit resolutions and complexes. Illusie gave the definition

of weak equivalence of simplicial presheaves which, in retrospect, leads

later to the idea of higher stack via the model categories of Jardine and


Joyal. Somewhere in these works is the idea, which seems to have been

communicated to Illusie by Deligne, of looking at the derived category

of diagrams as a functor of the base category; this was later taken up by

Grothendieck and Cisinski under the name “derivator” [109] [70].

In 1980, Dwyer and Kan came out with their theory of simplicial lo-

calization, allowing the association of a simplicial category to any pair

(M ,W ) and giving the higher categorical version of Gabriel-Zisman’s

theory. They developped an extensive theory of simplicial categories, in-

cluding several different constructions of the simplicial localization which

inverts the morphisms of W in a homotopical sense. This construction

provides the door passing from the world of categories to the world of

higher categories, because even if we start with a regular 1-category,

then localize by inverting a collection of morphisms, the simplicial lo-

calization is in general a simplicial category which is not a 1-category.

The simplicial localization maps to the usual or Gabriel-Zisman local-

ization but the latter is only the 1-truncation. So, if we want to invert a

collection of morphisms in a “homotopically correct” way, we are forced

to introduce some kind of higher categorical structure, at the very least

the notion of simplicial category. Unfortunately, the importance of the

Dwyer-Kan construction doesn’t seem to have been generally noticed at

the time.

During this period, the category-theorists and particularly the Aus-

tralian school, were working on fully understanding the theory of strictly

associative n-categories and∞-categories. In a somewhat different direc-

tion, Loday introduced the notion of catn-group which was obtained by

iterating the internal category construction in a different way, allowing

categories of objects as well as of morphisms. Ronnie Brown worked on

various aspects of the problem of relating these structures to homotopy

theory: the strictly associative n-categories don’t model all homotopy

types (Brown-Higgins), whereas the catn-groups do (Brown-Loday).

A major turning point in the history of higher categories was Alexan-

dre Grothendieck’s famous manuscript Pursuing Stacks, which started

out as a collection of letters to different colleagues with many parts

crossed out and rewritten, the whole circulated in mimeographed form.

I was lucky to be able to consult a copy in the back room of the Prince-

ton math library, and later to obtain a copy from Jean Malgoire; a pub-

lished version edited by Georges Maltsiniotis should appear soon [108].

Grothendieck introduces the problem of defining a notion of weakly as-

sociative n-category, and points out that many areas of mathematics

could benefit from such a theory, explaining in particular how a theory


of higher stacks should provide the right kind of coefficient system for

higher nonabelian cohomology.

Grothendieck made important progress in investigating the topology

and category theory behind this question. He introduced the notion of

n-groupoid, an n-category in which all arrows are invertible up to equiva-

lences at the next higher level. He conjectured the existence of a Poincare

n-groupoid construction

Πn : Top → n-Gpd ⊂ n-Cat

where n-Gpd is the collection of weakly associative n-groupoids. He pos-

tulated that this functor should provide an equivalence of homotopy

theories between n-truncated spaces1 and n-groupoids.

In his search for algebraic models for homotopy types, Grothendieck

was inspired by one of the pioneering works in this direction, the notion

of Catn-groups of Brown and Loday. This is what is now known as the

“cubical” approach where the set of objects can itself have a structure

for example of n− 1-category, so it isn’t quite the same as the approach

we are looking for, commonly called the “globular” case.2

Much of “Pursuing stacks” is devoted to the more general question

of modeling homotopy types by algebraic objects such as presheaves on

a fixed small category, developing a theory of “test categories” which

has now blossomed into a distinct subject in its own right thanks to the

further work of Maltsiniotis [160] and Cisinski [68]. One of the questions

their theory aims to address is, which presheaf categories provide good

models for homotopy theory. One could ask a similar question with re-

spect to Segal’s utilisation of ∆, namely whether other categories could

be used instead. We don’t currently have any good information about

this. As a start, throughout the book we will try to point out in discus-

sion and counterexamples the main places where special properties of ∆

are used.

In the parts of “Pursuing stacks” about n-categories, the following

theme emerges: the notion of n-category with strictly associative com-

position, is not sufficient. This is seen from the fact that strictly asso-

ciative n-categories satisfying a weak groupoid condition, do not serve

to model homotopy n-types as would be expected. Fundamentally due

1 A space T is n-truncated if πi(T, t) = 0 for all i > n and all basepoints t ∈ T .The n-truncated spaces are the objects which appear in the Postnikov tower offibrations, and one can define the truncation T → τ≤n(T ) for any space T , byadding cells of dimension ≥ n+ 2 to kill off the higher homotopy groups.

2 Paoli has recently defined a notion of special Catn-group [170] that imposes theglobularity condition weakly.


to Godement and the Eckmann-Hilton argument, this observation was

refined over time by Brown and Higgins [58] and Berger [29]. We discuss

it in some detail in Chapter 4.

Since strict n-categories aren’t enough, it leads to the question of

defining a notion of weak n-category, which is the main subject of our

book. Thanks to a careful reading by Georges Maltsiniotis, we now know

that Grothendieck’s manuscript in fact contained a definition of weakly

associative n-groupoid [161], and that his definition is very similar to

Batanin’s definition of n-category [162]. Grothendieck enunciated the

deceptively simple rule [108]:

Intuitively, it means that whenever we have two ways of associating to a finitefamily (ui)i∈I of objects of an ∞-groupoid, ui ∈ Fn(i), subjected to a standardset of relations on the ui’s, an element of some Fn, in terms of the ∞-groupoidstructure only, then we have automatically a “homotopy” between these builtin in the very structure of the ∞-groupoid, provided it makes sense to ask forone . . .

The structure of this as a definition was not immediately evident upon

any initial reading, all the more so when one takes into account the

directionality of arrows, so “Pursuing stacks” left open the problem of

giving a good definition of weakly associative n-category.

Given the idea that an equivalence Πn between homotopy n-types

and n-groupoids should exist, it becomes possible to think of replac-

ing the notion of n-groupoid by the notion of n-truncated space. This

motivated Joyal to define a closed model structure on the category of

simplicial sheaves, and Jardine to extend this to simplicial presheaves.

These theories give an approach to the notion of∞-stack, and were used

by Thomason, Voevodsky, Morel and others in K-theory.

Also explicitly mentioned in “Pursuing stacks” was the limiting case

n = ω, involving i-morphisms of all degrees 0 ≤ i < ∞. Again, an ω-

groupoid should correspond, via the inverse of a Poincare construction

Πω, to a full homotopy type up to weak equivalence.

We can now get back to the discussion of simplicial categories. These

are categories enriched over spaces, and applying Πω (which is supposed

to be compatible with products) to the morphism spaces, we can think of

simplicial categories as being categories enriched over ω-groupoids. Such

a thing is itself an ω-category A, with the property that the morphism

ω-categories A(x, y) are groupoids. In other words, the i-morphisms are

invertible for i ≥ 2, but not necessarily for i = 1. Jacob Lurie introduced

the terminology (∞, 1)-categories for these things, where more generally

an (∞, n)-category would be an ω-category such that the i-morphisms


are invertible up to equivalence, for i > n. The point of this discussion—

of notions which have not yet been defined—is to say that the notion

of simplicial category is a perfectly good substitute for the notion of

(∞, 1)-category even if we don’t know what an ω-category is in general.

This replacement no longer works if we want to look at n-categories

with noninvertible morphisms at levels ≥ 2, or somewhat similarly,

(∞, n)-categories for n ≥ 2. Grothendieck doesn’t seem to have been

aware of Dwyer and Kan’s work, just prior to “Pursuing stacks”, on

simplicial categories;3 however he was well aware that the notions of

n-category for small values of n had been extensively investigated ear-

lier in Benabou’s book about 2-categories [28], and Gordon, Powers and

Street on 3-categories [104]. The combinatorial explosion inherent in

these explicit theories was why Grothendieck asked for a different form

of definition which could work in general.

As he forsaw in a vivid passage [108, First letter, p. 16], there are cur-

rently many different definitions of n-category. This started with Street’s

proposal in [201], of a definition of weak n-category as a simplicial set

satisfying a certain variant of the Kan condition where one takes into

account the directions of arrows, and also using the idea of “thinness”.

His suggestion, in retrospect undoubtedly somewhat similar to Joyal’s

iteration of the notion of quasicategory, wasn’t worked out at the time,

but has recieved renewed interest, see Verity [212] for example.

The Segal-style approach to weak topological categories was intro-

duced by Dwyer, Kan, Smith [92] and Schwanzl, Vogt [184], but the fact

that they immediately proved a rectification result relating Segal cat-

egories back to strict simplicial categories, seems to have slowed down

their further consideration of this idea. Applying Segal’s idea seems to

have been the topic of a letter from Breen to Grothendieck in 1975, see

page 71 below.

Kapranov and Voevodsky in [136] considered a notion of “Poincare

∞-groupoid” which is a strictly associative ∞-groupoid but where the

arrows are invertible only up to equivalence. It now appears likely that

their constructions should best be interpreted using some kind of weak

unit condition [140].

At around the same time in the mid-1990’s, three distinct approaches

to defining weak n-categories came out: Baez and Dolan’s approach used

opetopes [7] [9], Tamsamani’s approach used iteration of the Segal de-

looping machine [205] [206], and Batanin’s approach used globular oper-

3 Paradoxically, Grothendieck’s unpublished manuscript is responsible in largepart for the regain of interest in Dwyer and Kan’s published papers!


ads [20] [21]. The Baez-Dolan and Batanin approaches will be discussed

in Chapter 6.

The work of Baez and Dolan was motivated by a far-reaching program

of conjectures on the relationship between n-categories and physics [6]

[10], which has led to important developments most notably the recent

proof by Hopkins and Lurie.

In relationship with Grothendieck’s manuscript, as we pointed out

above, Batanin’s approach is the one which most closely resembles what

Grothendieck was asking for, indeed Maltsiniotis generalized the defini-

tion of n-groupoid which he found in “Pursuing stacks”, to a definition

of n-category which is similar to Batanin’s one [162].

In the subsequent period, a number of other definitions have appeared,

and people have begun working more seriously on the approach which

had been suggested by Street. Batanin, in mentioning the letter from

Baez and Dolan to Street [7], also points out that Hermida, Makkai and

Power have worked on the opetopic ideas, leading to [114]. M. Rosellen

suggested in 1996 to give a version of the Segal-style definition, using the

theory of operads. He didn’t concretize this but Trimble gave a definition

along these lines, now playing an important role in work of Cheng [65].

Further ideas include those of Penon, Leinster’s multicategories, and

others. Tom Leinster has collected together ten different definitions in

the useful compendium [148]. The somewhat mysterious [143] could also

be pointed out. In the simplicial direction Rezk’s complete Segal spaces

[178] can be iterated as suggested by Barwick [179], and Joyal proposes

an iteration of the method of quasicategories [127].

We shall discuss the simplicial definitions in Chapter 5 and the op-

eradic definitions in Chapter 6. One of the main tasks in the future will

be to understand the relationships between all of these approaches. Our

goal here is more down-to-earth: we would like to develop the tools nec-

essary for working with Tamsamani’s n-categories. We hope that similar

tools can be developped for the other approaches, making an eventual

comparison theory into a powerful method whereby the particular ad-

vantages of each definition could all be put in play at the same time.

Tamsamani defined the Poincare n-groupoid functor for his notion

of n-category, and showed Grothendieck’s conjectured equivalence with

the theory of homotopy n-types [206]. The same has also been done for

Batanin’s theory, by Berger in [30].

It is interesting to note that the two main ingredients in Tamsamani’s

approach, the multisimplicial nerve construction and Segal’s deloop-

ing machine, are both mentioned in “Pursuing stacks”. In particular,


Grothendieck reproduces a letter from himself to Breen dated July 1975,

in which Grothendieck acknowledges having recieved a proposed defi-

nition of non-strict n-category from Breen, a definition which accord-

ing to loc. cit “...has certainly the merit of existing...”. It is not clear

whether this proposed construction was ever worked out. Quite appar-

ently, Breen’s suggestion for using Segal’s delooping machine must have

gone along the lines of what we are doing here. Rather than taking

up this direction, Grothendieck elaborated a general ansatz whereby

n-categories would have various different composition operations, and

natural equivalences between any two natural compositions with the

same source and target, an idea now fully developed in the context of

Batanin’s and related definitions.

Once one or more points of view for defining n-categories are in hand,

the main problem which needs to be considered is to obtain—hopefully

within the same point of view—an n+1-category nCAT parametrizing

the n-categories of that point of view. This problem, already clearly

posed in “Pursuing stacks”, is one of our main goals in the more technical

central part of the book, for one model.

It turns out that Quillen’s technique of model categories, subsequently

deepened by several generations of mathematicians, is a great way of

attacking this problem. It is by now well-known that closed model cat-

egories provide an excellent environment for studying homotopy the-

ory, as became apparent from the work of Bousfield, Dwyer and Kan

on closed model categories of diagrams, and the generalization of these

ideas by Joyal, Jardine, Thomason and Voevodsky who used model cat-

egories to study simplicial presheaves under Illusie’s condition of weak

equivalence. In the Segal-style paradigm of weak enrichment, we look at

functors ∆o → (n − 1)Cat, so we are certainly also studying diagrams

and it is reasonable to expect the notion of model category to bring some

of the same benefits as for the above-mentioned theories.

To be more precise about this motivation, recall from “Pursuing stacks”

that nCAT should be an n+1-category whose objects are in one-to-one

correspondence with the n-categories of a given universe. The structure

of n + 1-category therefore consists of specifying the morphism objects

HomnCAT (A,B) which should themselves be n-categories parametrizing

“functors” (in an appropriate sense) from A to B.

In the explicit theories for n = 2, 3, 4 this is one of the places where

a combinatorial explosion takes place: the functors from A to B have to

be taken in a weak sense, that is to say we need a natural equivalence


between the image of a composition and the composition of the images,

together with the appropriated coherence data at all levels.

The following simple example shows that, even if we were to consider

only strict n-categories, the strict morphisms are not enough. Suppose

G is a group and V an abelian group and we set A equal to the category

with one object and group of automorphisms G, and B equal to the

strict n-category with only one i-morphism for i < n and group V of

n-automorphisms of the unique n − 1-morphism; then for n = 1 the

equivalence classes of strict morphisms from A to B are the elements of

H1(G, V ) so we would expect to get Hn(G, V ) in general, but for n > 1

there are no nontrivial strict morphisms from A to B. So some kind of

weak notion of functor is needed.

Here is where the notion of model category comes in: one can view

this situation as being similar to the problem that usual maps between

simplicial sets are generally too rigid and don’t reflect the homotopical

maps between spaces. Kan’s fibrancy condition and Quillen’s formaliza-

tion of this in the notion of model category, provide the solution: we

should require the target object to be fibrant and the source object to

be cofibrant in an appropriate model category structure. Quillen’s ax-

ioms serve to guarantee that the notions of cofibrancy and fibrancy go

together in the right way. So, in the application to n-categories we would

like to define a model structure and then say that the usual maps A → B

strictly respecting the structure, are the right ones provided that A is

cofibrant and B fibrant.

To obtain nCAT a further property is needed, indeed we are not just

looking to find the right maps from A to B but to get a morphism object

HomnCAT (A,B) which should itself be an n-category. It is natural to

apply the idea of “internal Hom”, that is to put

HomnCAT (A,B) := Hom(A,B)

using an internal Hom in our model category. For our purposes, it is

sufficient to consider Hom adjoint to the direct product operation, in

other words a map

E → Hom(A,B)

should be the same thing as a map E × A → B. This obviously implies

imposing further axioms on the closed model structure, in particular

compatibility between × and cofibrancy since the direct product is used

on the source side of the map. It turns out that the required axioms

are already well-known in the notion of monoidal model category [120],


which is a model category provided with an additional operation ⊗, and

certain axioms of compatibility with the cofibrant objects. In our case,

the operation is already given as the direct product ⊗ = × of the model

category, and a model category which is monoidal for the direct product

operation will be called cartesian (Chapter 10).

In the present book, we are concentrating on Tamsamani’s approach to

n-categories, which in [117] was modified to “ Segal n-categories” in the

course of discussions with Andre Hirschowitz. In Tamsamani’s theory an

n-category is viewed as a category enriched over n− 1-categories, using

Segal’s machine to deal with the enrichment in a homotopically weak

way.

In Regis Pelissier’s thesis, following a question posed by Hirschowitz,

this idea was pushed to a next level: to study weak Segal-style enrichment

over a more general model category, with the aim of making the iteration

formal. A small link was missing in this process at the end of [171],

essentially because of an error in [193] which Pelissier discovered. He

provided the correction when the iterative procedure is applied to the

model category of simplicial sets. But in fact, his patch applies much

more generally if we just consider the operation of functoriality under

change of model categories.

This is what we will be doing here. But instead of following Pelissier’s

argument too closely, some aspects will be set into a more general discus-

sion of certain kinds of left Bousfield localizations. The idea of breaking

down the construction into several steps including a main step of left

Bousfield localization, is due to Clark Barwick.

The Segal 1-categories are, as was originally proven in [92], equivalent

to strict simplicial categories. Bergner has shown that this equivalence

takes the form of a Quillen equivalence between model categories [34].

However, the model category of simplicial categories is not appropriate

for the considerations described above: it is not cartesian, indeed the

product of two cofibrant simplicial categories will not be cofibrant.4 It

is interesting to imagine several possible ways around this problem: one

could try to systematically apply the cofibrant replacement operation;

this would seem to lead to a theory very similar to the consideration of

Gray tensor products of Leroy [150] and Crans [76]; or one could hope for

a general construction replacing a model category by a cartesian one (or

perhaps, given a model category with monoidal structure incompatible

4 This remark also applies to the projective model structure for weakly enrichedSegal-style categories, whereas on the other hand the projective structure ismuch more practical for calculating maps.


with cofibrations, construct a monoidal model category in some sense

equivalent to it).

As Bergner pointed out, the theories of simplicial categories and Segal

categories are also equivalent to Charles Rezk’s theory of complete Segal

spaces. As we shall discuss further in Chapter 5, Rezk requires that

the Segal maps be weak equivalences, but rather than having A0 be a

discrete simplicial set corresponding to the set of objects, he asks that

A0 be a simplicial set weakly equivalent to the “interior” Segal groupoid

of A. Barwick pointed out that Rezk’s theory could also be iterated, and

Rezk’s recent preprint [179] shows that the resulting model category is

cartesian. So, this route also leads to a construction of nCAT and can

serve as an alternative to what we are doing here. It should be possible

to extend Bergner’s comparison result to obtain equivalences between

the iterates of Rezk’s theory and the iterates we consider here. If our

current theory is perhaps simpler in its treatment of the set of objects,

Rezk’s theory has a better behavior with respect to homotopy limits.

As more different points of view on higher categories are up and run-

ning, the comparison problem will be posed: to find an appropriate way

to compare different points of view on n-categories and (one hopes) to

say that the various points of view are equivalent and in particular that

the various n+1-categories nCAT are equivalent via these comparisons.

Grothendieck gave a vivid description of this problem (with remarkable

foresight, it would seem [148]) in the first letter of [108]. He pointed out

that it is not actually clear what type of general setup one should use for

such a comparison theory. Various possibilities would include the model

category formalism, or the formalism of (∞, 1)-categories starting with

Dwyer-Kan localization and moving through Lurie’s theory.

Within the domain of simplicial theories, we have mentioned Bergner’s

comparison between three different approaches to (∞, 1)-categories [34].

A further comparison of these theories with quasicategories is to be found

in Lurie [153].

A recent result due to Cheng [65] gives a comparison between Trim-

ble’s definition and Batanin’s definition (with some modifications on

both sides due to Cheng and Leinster). Batanin’s approach used op-

erads more as a way of encoding general algebraic structures, and is

the closest to Grothendieck’s original philosophy. While also operadic,

Trimble’s definition is much closer to the philosophy we are developing

in the present book, whereby one goes from topologists’ delooping ma-

chinery (in his case, operads) to an iterative theory of n-categories. It is

to be hoped that Cheng’s result can be expanded in various directions to


obtain comparisons between a wide range of theories, maybe using Trim-

ble’s definition as a bridge towards the simplicial theories. This should

clearly be pursued in the near future, but it would go beyond the scope

of the present work.

We now turn to the question of potential applications. Having a good

theory of n-categories should open up the possibility to pursue any of

the several programs such as that outlined by Grothendieck [108], the

generalization to n-stacks and n-gerbs of the work of Breen [51], or the

program of Baez and Dolan in topological quantum field theory [6].

Once the theory of n-stacks is off the ground this will give an algebraic

approach to the “geometric n-stacks” considered in [192].

As the title indicates, Grothendieck’s manuscript was intended to de-

velop a foundational framework for the theory of higher stacks. In turn,

higher stacks should be the natural coefficients for nonabelian cohomol-

ogy, the idea being to generalize Giraud’s [102] to n ≥ 3.

The example of diagrams of spaces translates, via the construction

Πn, to a notion of diagram of n-groupoids. This is a strict version of

the notion of n-prestack in groupoids which would be a weak functor

from the base category Ψ to the n+ 1-category GPDn of n-groupoids.

Grothendieck introduced the notion of n-stack which generalizes to n-

categories the classical notion of stack. A full discussion of this theory

would go beyond the scope of the present work: we are just trying to set

up the n-categorical foundations first. The notion of n-stack, maybe with

n =∞, has applications in many areas as predicted by Grothendieck.

Going backwards along Πn, it turns out that diagrams of spaces or

equivalently simplicial presheaves, serve as a very adequate replacement

[125] [123] [209] [213] [168]. So, the notion of n-categories as a prereq-

uisite for higher stacks has proven somewhat illusory. And in fact, the

model category theory developed for simplicial presheaves has been use-

ful for attacking the theory of n-categories as we do here, and also for

going from a theory of n-categories to a theory of n-stacks, as Hollander

has done for 1-stacks [118] and as Hirschowitz and I did for n-stacks in

[117].

An n-stack on a site X will be a morphism X → nCAT . This

requires a construction for the n+ 1-category nCAT , together with the

appropriate notion of morphism between n + 1-categories. The latter

is almost equivalent to knowing how to construct the n + 2-category

(n + 1)CAT of n + 1-categories. From this discussion the need for an

iterative approach to the theory of n-categories becomes clear.


My own favorite application of stacks is that they lead in turn to a

notion of nonabelian cohomology. Grothendieck says [108]:

Thus n-stacks, relativized over a topos to “n-stacks over X”, are viewed pri-marily as the natural “coefficients” in order to do (co)homological algebra ofdimension ≤ n over X.

The idea of using higher categories for nonabelian cohomology goes

back to Giraud [102], and had been extended to the cases n = 2, 3

by Breen somewhat more recently [49]. Breen’s book motivated us to

proceed to the case of n-categories at the beginning of Tamsamani’s

thesis work.

Another utilisation of the notion of n-category is to model homotopy

types. For this to be useful one would like to have as simple and compact

a definition as possible, but also one which lends itself to calculation.

The simplicial approach developped here is direct, but it is possible that

the operadic approaches which will be mentioned in Chapter 6 could be

more amenable to topological computations. An iteration of the classical

Segal delooping machine has been considered by Dunn [85].

The Poincare n-groupoid of a space is a generalization of the Poincare

groupid Π1(X), a basepoint-free version of the fundamental group π1(X)

popularized by Ronnie Brown [55]. Van Kampen’s theorem allows for

computations of fundamental groups, and as Brown has often pointed

out, it takes a particularly nice form when written in terms of the

Poincare groupoid: it says that if a space X is written as a pushout

X = U ∪W V then the Poincare groupoid is a pushout in the 2-category

of groupoids:

Π1(X) = Π1(U) ∪Π1(W ) Π1(V ).

This says that Π1 commutes with colimits.

Extending this theory to the case of Poincare n-groupoids is one of

the motivations for introducing colimits and indeed the whole model-

categoric machinery for n-categories. We will then be able to write, in

case of a pushout of spaces X = U ∪W V ,

Πn(X) = Πn(U) ∪Πn(W ) Πn(V ).

Of course the pushout diagram of spaces should satisfy some excision

condition as in the original Van Kampen theorem, and this may be

abstracted by refering to simplicial sets instead.

The homotopy theory and nonabelian cohomology motivations may

be combined by looking for a higher-categorical theory of shape. For a


space X we can define the nonabelian cohomology n-category H(X,A)

with coefficients in an n-stack F over X . This applies in particular to

the constant stack AX associated to an n-category A. The functor

A 7→ H(X,AX)

is co-represented by the universal element

ηX ∈ H(X,Πn(X)X),

giving a way of characterizing Πn(X) by universal property. This essen-

tially tautological observation paves the way for more nontrivial shape

theories, consisting of an n-category COEFF and a functor Shape(X) :

COEFF → COEFF . A particularly useful version is when COEFF

is the n-category of certain n-stacks over a site Y , and Shape(X)(F ) =

Hom(Πn(X)Y ,F ) where Πn(X)Y denotes the constant stack on Y

with values equal to Πn(X). This leads to subjects generalizing Mal-

cev completions and rational homotopy theory [110] [111] [112], such as

the schematization of homotopy types [210] [137] [174] [169], de Rham

shapes and nonabelian Hodge theory.

One of the main advantages to a theory of higher categories, is that

the notions of homotopy limit and homotopy colimit, by now classical in

algebraic topology, become internal notions in a higher category. Indeed,

they become direct analogues of the notions of limit and colimit in a

usual 1-category, with corresponding universal properties and so on. This

has an interesting application to the “abelian” case: the structure of

triangulated category is automatic once we know the (∞, 1)-categorical

structure. This was pointed out by Bondal and Kapranov in the dg

setting [43]: their enhanced triangulated categories are just dg-categories

satisfying some further axioms; the structure of triangles comes from the

dg structure. Historically one can trace this observation back to the end

of Gabriel-Zisman’s book [100], although nobody seems to have noticed

it until rediscovered by Bondal and Kapranov.

I first learned of the notion of “2-limit” from the paper of Deligne and

Mumford [81], where it appears at the beginning with very little expla-

nation (their paper should also be added to the list of motivations for

developing the theory of higher stacks). Several authors have since con-

sidered 2-limits and 2-topoi, originating with Bourn [45] and continuing

recently with Weber [214] for example.

The notions of homotopy limits and colimits internalized in an (∞, 1)-

category have now recieved an important foundational formulation with

Lurie’s work on ∞-topoi [153].


Power has given an extensive discussion of the motivations for higher

categories stemming from logic and computer science, in [173]. He points

out the role played by weak limits. Recently, Gaucher, Grandis and oth-

ers have used higher categorical notions to study directed and concurrent

processes [101] [105]. It would be interesting to see how these theories

interact with the notion of ∞-topos.

Recall that gerbs played an important role in descent theory and non-

neutral tannakian categories [82]. Current developments where the no-

tion of higher category is more or less essential on a foundational level,

include “derived algebraic geometry” and higher tannakian theory. It

would go beyond our present scope to discuss these here but the reader

may search for numerous references.

Stacks and particular gerbes of higher groupoids have found many

interesting applications in the mathematical physics literature, starting

with explicit considerations for 1- and 2-gerbes. Unfortunately it would

go beyond our scope to list all of these. However, one of the main con-

tributions from mathematical physics has been to highlight the utility

of higher categories which are not groupoids, in which there can be

non-invertible morphisms. Explicit first cases come about when we con-

sider monoidal categories: they may be considered as 2-categories with

a single object. And then braided monoidal categories may be consid-

ered as 3-categories with a single object and a single 1-morphism, where

the braiding isomorphism comes from the Eckmann-Hilton argument.

These entered into the vast program of research on combinatorial quan-

tum field theories and knot invariants—again the reader is left to fill in

the references here.

John Baez and Jim Dolan provided a major impetus to the theory

of higher categories, by formulating a series of conjectures about how

the known relationships between low-dimensional field theories and n-

categories for small values of n, should generalize in all dimensions [6] [8]

[9] [10]. On the topological or field-theoretical side, they conjecture the

existence of a k-fold monoidal n-category (or equivalently, a k-connected

n+ k-category) representing k-dimensional manifolds up to cobordism,

where the higher morphisms should correspond to manifolds with cor-

ners. On the n-categorical side, they propose a notion of n-category with

duals in which all morphisms should have internal adjoints. Then, their

main conjecture relating these two sides is that the cobordism n + k-

category should be the universal n + k-category with duals generated

by a single morphism in degree k. The specification of a field theory is

a functor from this n+ k-category to some other one, and it suffices to


specify the image of the single generating morphism. They furthermore

go on to investigate possible candidates for the target categories of such

functors, looking at higher Hilbert spaces and other such things. We will

include some discussion (based on [196]) of one of Baez and Dolan’s pre-

liminary conjectures, the stabilization hypothesis, in an ulterior version

of the present manuscript.

The Baez-Dolan conjectures step outside of the realm of n-groupoids,

so they really require an approach which can take into account non-

invertible morphisms. In their “n-categories with duals”, they gener-

alize the fact that the notion of adjoint functor can be expressed in

2-categorical terms within the 2-category 1CAT . Mackaay describes the

application of internal adjoints to 4-manifold invariants in [157]. The

notion of adjoint generalizes within an n-category to the notion of dual

of any i-morphism for 0 < i < n. At the top level of n-morphisms, the

dual operation should either be: ignored; imposed as additional struc-

ture; or pushed to∞ by considering directly the theory of∞-categories.

Of course, a morphism which is really invertible is automatically dualiz-

able and its dual is the same as its inverse, so the interesting n-categories

with duals have to be ones which are not n-groupoids.

As Cheng has pointed out [64], in the last case one obtains a structure

which looks algebraically like an∞-groupoid, so the distinction between

invertible and dualizable morphisms should probably be considered as

an additional more analytic structure in itself. We don’t yet have the

tools to fully investigate the theory of ∞-categories. Further comments

on these issues will be made in Section 5.7.

In a very recent development, Hopkins and Lurie have announced a

proof of a major part of the Baez-Dolan conjectures, saying that the

category of manifolds with appropriate corners, and cobordisms as i-

morphisms, is the universal n-category with duals generated by a single

element. This universal property allows one to define a functor from

the cobordism n-category to any other n-category with duals, by simply

specifying a single object. I hope that some of the techniques presented

here can help in understanding this fascinating subject.

2

Strict n-categories

Classically, the first and easiest notion of higher category was that of

strict n-category. We review here some basic definitions, as they intro-

duce important notions for weak n-categories. In Chapter 4 we will point

out why the strict theory is generally considered not to be sufficient.

In the current chapter only, all n-categories are meant to be strict

n-categories. For this reason we try to put in the adjective “strict” as

much as possible when n > 1; but in any case, the very few times that

we speak of weak n-categories, this will be explicitly stated. We mostly

restrict our attention to n ≤ 3.

In case that isn’t already clear, it should be stressed that everything

we do in this section (as well as most of the next and even the subsequent

one as well) is very well known and classical, so much so that I don’t

know what are the original references.

To start with, a strict 2-category A is a collection of objects A0 plus,

for each pair of objects x, y ∈ A0 a category A(x, y) together with a

morphism

A(x, y) ×A(y, z) → A(x, z)

which is strictly associative in the obvious way; and such that a unit

exists, that is an element 1x ∈ Ob(A(x, x)) with the property that mul-

tiplication by 1x acts trivially on objects of A(x, y) or A(y, x) and mul-

tiplication by 11x acts trivially on morphisms of these categories.

A strict 3-category C is the same as above but where C(x, y) are sup-

posed to be strict 2-categories. There is an obvious notion of direct

product of strict 2-categories, so the above definition applies mutatis

mutandis.

For general n, the well-known definition is most easily presented by

induction on n. We assume known the definition of strict n− 1-category


30 Strict n-categories

for n−1, and we assume known that the category of strict n−1-categories

is closed under direct product. A strict n-category C is then a category

enriched [139] over the category of strict n − 1-categories. This means

that C is composed of a set of objects Ob(C) together with, for each pair

x, y ∈ Ob(C), a morphism-object C(x, y) which is a strict n− 1-category;

together with a strictly associative composition law

C(x, y)× C(y, z) → C(x, z)

and a morphism 1x : ∗ → C(x, x) (where ∗ denotes the final object cf

below) acting as the identity for the composition law. The category of

strict n-categories denoted nStrCat is the category whose objects are as

above and whose morphisms are the transformations strictly perserving

all of the structures. Note that nStrCat admits a direct product: if C

and C′ are two strict n-categories then C × C′ is the strict n-category

with

Ob(C × C′) := Ob(C)×Ob(C′)

and for (x, x′), (y, y′) ∈ Ob(C × C′),

(C × C′)((x, x′), (y, y′)) := C(x, y)× C′(x′, y′)

where the direct product on the right is that of (n−1)StrCat. Note that

the final object of nStrCat is the strict n-category ∗ with exactly one

object x and with ∗(x, x) = ∗ being the final object of (n− 1)StrCat.

The induction inherent in this definition may be worked out explicitly

to give the definition as it is presented in [136] for example. In doing

this one finds that underlying a strict n-category C are the sets Mori(C)

of i-morphisms or i-arrows, for 0 ≤ i ≤ n. The set of 0-morphisms is

by definition the set of objects Mor0(C) := Ob(C), and Mori(C) is the

disjoint union over all pairs x, y of the Mori−1(C(x, y)). These fit together

in a diagram called a (reflexive) globular set:

· · ·Mori+1(C)s→←t→ Mori(C)

s→←t→ Mori−1(C) · · ·Mor1(C)

s→←t→ Mor0(C)

where the rightward maps are the source and target maps and the left-

ward maps are the identity maps1. These may be defined inductively

using the definition we have given of Mori(C). The structure of strict n-

category on this underlying globular set is determined by further com-

position laws at each stage: the i-morphisms may be composed with

1 The adjective “reflexive” refers to the inclusion of these leftward “identity”maps; a non-reflexive globular set would have only the s and t.

2.1 Godement, Interchange or the Eckmann-Hilton argument 31

respect to the j-morphisms for any 0 ≤ j < i, operations denoted in

[136] by ∗j . These are partially defined depending on iterations of the

source and target maps. For a more detailed explanation, refer to the

standard references including [58] [201] [136] [28] [100].

2.1 Godement, Interchange or the Eckmann-Hilton

argument

One of the most important of the axioms satisfied by the various com-

positions in a strict n-category is variously known under the name of

“Eckmann-Hilton argument”, “Godement relations”, “interchange rules”

etc. This property comes from the fact that the composition law

C(x, y)× C(y, z) → C(x, z)

is a morphism with domain the direct product of the two morphism n−1-

categories from x to y and from y to z. In a direct product, compositions

in the two factors by definition are independent (commute). Thus, for

1-morphisms in C(x, y) × C(y, z) (where the composition ∗0 for these

n − 1-categories is actually the composition ∗1 for C and we adopt the

latter notation), we have

(a, b) ∗1 (c, d) = (a ∗1 c, b ∗1 d).

This leads to the formula

(a ∗0 b) ∗1 (c ∗0 d) = (a ∗1 c) ∗0 (b ∗1 d).

This seemingly innocuous formula takes on a special meaning when we

start inserting identity maps. Suppose x = y = z and let 1x be the

identity of x which may be thought of as an object of C(x, x). Let e

denote the 2-morphism of C, identity of 1x; which may be thought of as

a 1-morphism of C(x, x). It acts as the identity for both compositions

∗0 and ∗1 (the reader may check that this follows from the part of the

axioms for an n-category saying that the morphism 1x : ∗ → C(x, x) is

an identity for the composition).

If a, b are also endomorphisms of 1x, then the above rule specializes

to:

a ∗1 b = (a ∗0 e) ∗1 (e ∗0 b) = (a ∗1 e) ∗0 (e ∗1 b) = a ∗0 b.

Thus in this case the compositions ∗0 and ∗1 are the same. A different


ordering gives the formula

a ∗1 b = (e ∗0 a) ∗1 (b ∗0 e) = (e ∗1 b) ∗0 (a ∗1 e) = b ∗0 a.

Therefore we have

a ∗1 b = b ∗1 a = a ∗0 b = b ∗0 a.

This argument says, then, that Ob(C(x, x)(1x, 1x)) is a commutative

monoid and the two natural multiplications are the same.

The same argument extends to the whole monoid structure on the

n− 2-category C(x, x)(1x, 1x):

Lemma 2.1.1 The two composition laws on the strict n− 2-category

C(x, x)(1x, 1x) are equal, and this law is commutative. In other words,

C(x, x)(1x, 1x) is an abelian monoid-object in the category (n−2)StrCat.

There is a partial converse to the above observation: if the only object

is x and the only 1-morphism is 1x then nothing else can happen and

we get the following equivalence of categories.

Lemma 2.1.2 Suppose G is an abelian monoid-object in the category

(n− 2)StrCat. Then there is a unique strict n-category C such that

Ob(C) = x and Mor1(C) = Ob(C(x, x)) = 1x

and such that C(x, x)(1x, 1x) = G as an abelian monoid-object. This

construction establishes an equivalence between the categories of abelian

monoid-objects in (n−2)StrCat, and the strict n-categories having only

one object and one 1-morphism.

Proof: Define the strict n − 1-category U with Ob(U) = u and

U(u, u) = G with its monoid structure as composition law. The fact

that the composition law is commutative allows it to be used to define

an associative and commutative multiplication

U × U → U .

Now let C be the strict n-category with Ob(C) = x and C(x, x) = U

with the above multiplication. It is clear that this construction is inverse

to the previous one.

It is clear from the construction (the fact that the multiplication on U

is again commutative) that the construction can be iterated any number

of times. We obtain the following “delooping” corollary.


Corollary 2.1.3 Suppose C is a strict n-category with only one object

and only one 1-morphism. Then there exists a strict n + 1-category B

with only one object b and with B(b, b) ∼= C.

Proof: By the previous lemmas, C corresponds to an abelian monoid-

object G in (n − 2)StrCat. Construct U as in the proof of 2.1.2, and

note that U is an abelian monoid-object in (n − 1)StrCat. Now apply

the result of 2.1.2 directly to U to obtain B ∈ (n+1)StrCat, which will

have the desired property.

2.2 Strict n-groupoids

Recall that a groupoid is a category where all morphisms are invertible.

This definition generalizes to strict n-categories in the following way, as

was pointed out by Kapranov and Voevodsky [136]. We give a theorem

stating that three versions of this definition are equivalent (one of these

equivalences was left as an exercise in [136]).

Note that, following [136], we do not require strict invertibility of

morphisms, thus the notion of strict n-groupoid is more general than

the notion employed by Brown and Higgins [58].

Our discussion is in many ways parallel to the treatment of the groupoid

condition for weak n-categories [206] to be discussed in the next chapter,

and our treatment in this section comes in large part from discussions

with Tamsamani about this.

We state a few definitions and results which will then be proven all

together by induction on n.

Theorem 2.2.1 Fix an integer n ≥ 1. Suppose A is a strict n-category.

The following three conditions are equivalent (and in this case we say

that A is a strict n-groupoid).

(1) A is an n-groupoid in the sense of [136];

(2) for all x, y ∈ A, A(x, y) is a strict n − 1-groupoid, and for any

1-morphism f : x → y in A, the two morphisms of composition with f

A(y, z) → A(x, z), A(w, x) → A(w, y)

are equivalences of strict n− 1-groupoids (see below);

(3) for all x, y ∈ Ob(A), A(x, y) is a strict n − 1-groupoid, and the

truncation τ≤1A (defined in the next proposition) is a 1-groupoid.

Proposition 2.2.2 If A is a strict n-groupoid, then define τ≤kA to


be the strict k-category whose i-morphisms are those of A for i < k

and whose k-morphisms are the equivalence classes of k-morphisms of

A under the equivalence relation that two are equivalent if there is a k+1-

morphism joining them. The fact that this is an equivalence relation is

a statement about n − k-groupoids. The set τ≤0A will also be denoted

π0A. The truncation is again a k-groupoid, and for n-groupoids A the

truncation coincides with the operation defined in [136].

If A is a strict n-groupoid, define π0(A) := τ≤0(A). For x ∈ Ob(A)

define π1(A, x) := (τ≤1(A))(x, x), which is a group since τ≤1(A) is a

groupoid by the previous proposition. For 2 ≤ i ≤ n define by induction

πi(A, x) := πi−1(A(x, x), 1x). The interchange property allows to show

that this is an abelian group. These classical definitions are recalled in

[136].

Definition 2.2.3 A morphism f : A → B of strict n-groupoids is said

to be an equivalence if the following equivalent conditions are satisfied:

(a) f induces an isomorphism π0A → π0B, and for every object a ∈

Ob(A) f induces an isomorphism πi(A, a)∼=→ πi(B, f(a));

(b) f induces a surjection π0A → π0B and, for every pair of objects

x, y ∈ Ob(A), an equivalence of n−1-groupoids A(x, y) → B(f(x), f(y));

(c) if u, v are i-morphisms in A sharing the same source and target, and

if r is an i+1-morphism in B going from f(u) to f(v) then there exists

an i + 1-morphism t in A going from u to v and an i + 2-morphism in

B going from f(t) to r (this includes the limiting cases i = −1 where

u and v are not specified, and i = n − 1, n where “n + 1-morphisms”

mean equalities between n-morphisms and “n + 2-morphisms” are not

specified).

Lemma 2.2.4 If f : A → B and g : B → C are morphisms of strict

n-groupoids and if any two of f , g and gf are equivalences, then so is

the third. If

Af→ B

g→ C

h→ D

are morphisms of strict n-groupoids and if hg and gf are equivalences,

then g is an equivalence.

It is clear for n = 0, so we assume n ≥ 1 and proceed by induction on

n: we assume that Theorem 2.2.1 and the subsequent Proposition 2.2.2,

Definition 2.2.3, as well as Lemma 2.2.4, are known for strict n − 1-

categories.

We first discuss the existence of truncation (Proposition 2.2.2), for


k ≥ 1. Note that in this case τ≤kA may be defined as the strict k-

category with the same objects as A and with

(τ≤kA)(x, y) := τ≤k−1A(x, y).

Thus the fact that the relation in question is an equivalence relation, is

a statement about n− 1-categories and known by induction. Note that

the truncation operation clearly preserves any one of the three groupoid

conditions (1), (2), (3). Thus we may affirm in a strong sense that τ≤k(A)

is a k-groupoid without knowing the equivalence of the conditions (1)-

(3).

Note also that the truncation operation for n-groupoids is the same

as that defined in [136] (they define truncation for general strict n-

categories but for n-categories which are not groupoids, their definition

is different from that of [206] and not all that useful).

For 0 ≤ k ≤ k′ ≤ n we have

τ≤k(τ≤k′ (A)) = τ≤k(A).

To see this note that the equivalence relation used to define the k-arrows

of τ≤k(A) is the same if taken in A or in τ≤k+1(A)—the existence of a

k+1-arrow going between two k-arrows is equivalent to the existence of

an equivalence class of k + 1-arrows going between the two k-arrows.

Finally using the above remark we obtain the existence of the trunca-

tion τ≤0(A): the relation is the same as for the truncation τ≤0(τ≤1(A)),

and τ≤1(A) is a strict 1-groupoid in the usual sense so the arrows are

invertible, which shows that the relation used to define the 0-arrows (i.e.

objects) in τ≤0(A) is in fact an equivalence relation.

We complete our discussion of truncation by noting that there is a nat-

ural morphism of strict n-categories A → τ≤k(A), where the right hand

side (a priori a strict k-category) is considered as a strict n-category in

the obvious way.

We turn next to the notion of equivalence (Definition 2.2.3), and prove

that conditions (a) and (b) are equivalent. This notion for n-groupoids

will not enter into the subsequent treatment of Theorem 2.2.1—what

does enter is the notion of equivalence for n − 1-groupoids, which is

known by induction—so we may assume the equivalence of definitions

(1)-(3) for our discussion of Definition 2.2.3.

Recall first of all the definition of the homotopy groups. Let 1ia denote

the i-fold iterated identity of an object a; it is an i-morphism, the identity


of 1i−1a (starting with 10a = a). Then

πi(A, a) := τ≤i(A)(1i−1a , 1i−1

a ).

This definition is completed by setting π0(A) := τ≤0(A). These defini-

tions are the same as in [136]. Note directly from the definition that for

i ≤ k the truncation morphism induces isomorphisms

πi(A, a)∼=→ πi(τ≤k(A), a).

Also for i ≥ 1 we have

πi(A, a) = πi−1(A(a, a), 1a).

One shows that the πi are abelian for i ≥ 2. This is a consequence of

the Eckmann-Hilton argument discussed in the previous section.

Suppose f : A → B is a morphism of strict n-groupoids satisfying

condition (b). From the immediately preceding formula and the induc-

tive statement for n− 1-groupoids, we get that f induces isomorphisms

on the πi for i ≥ 1. On the other hand, the truncation τ≤1(f) satisfies

condition (b) for a morphism of 1-groupoids, and this is readily seen to

imply that π0(f) is an isomorphism. Thus f satisfies condition (a).

Suppose on the other hand that f : A → B is a morphism of strict

n-groupoids satisfying condition (a). Then of course π0(f) is surjective.

Consider two objects x, y ∈ A and look at the induced morphism

fx,y : A(x, y) → B(f(x), f(y)).

We claim that fx,y satisfies condition (a) for a morphism of n − 1-

groupoids. For this, consider a 1-morphism from x to y, i.e. an object r ∈

A(x, y). By version (2) of the groupoid condition for A, multiplication

by r induces an equivalence of n− 1-groupoids

m(r) : A(x, x) → A(x, y),

and furthermore m(r)(1x) = r. The same is true in B: multiplication by

f(r) induces an equivalence

m(f(r)) : B(f(x), f(x)) → B(f(x), f(y)).

The fact that f is a morphism implies that these fit into a commutative


square

A(x, x) → A(x, y)

B(f(x), f(x))

↓

→ B(f(x), f(y)).

↓

The equivalence condition (a) for f implies that the left vertical mor-

phism induces isomorphisms

πi(A(x, x), 1x)∼=→ πi(B(f(x), f(x)), 1f(x)).

Therefore the right vertical morphism (i.e. fx,y) induces isomorphisms

πi(A(x, y), r)∼=→ πi(B(f(x), f(y)), f(r)),

this for all i ≥ 1. We have now verified these isomorphisms for any

base-object r. A similar argument implies that fx,y induces an injection

on π0. On the other hand, the fact that f induces an isomorphism on

π0 implies that fx,y induces a surjection on π0 (note that these last two

statements are reduced to statements about 1-groupoids by applying τ≤1

so we don’t give further details). All of these statements taken together

imply that fx,y satisfies condition (a), and by the inductive statement of

the theorem for n− 1-groupoids this implies that fx,y is an equivalence.

Thus f satisfies condition (b).

We now remark that condition (b) is equivalent to condition (c) for

a morphism f : A → B. Indeed, the part of condition (c) for i = −1

is, by the definition of π0, identical to the condition that f induces

a surjection π0(A) → π0(B). And the remaining conditions for i =

0, . . . , n + 1 are identical to the conditions of (c) corresponding to j =

i − 1 = −1, . . . , (n − 1) + 1 for all the morphisms of n − 1-groupoids

A(x, y) → B(f(x), f(y)). (In terms of u and v appearing in the condition

in question, take x to be the source of the source of the source . . . ,

and take y to be the target of the target of the target . . . ). Thus by

induction on n (i.e. by the equivalence (b)⇔ (c) for n−1-groupoids), the

conditions (c) for f for i = 0, . . . , n+1, are equivalent to the conditions

that A(x, y) → B(f(x), f(y)) be equivalences of n− 1-groupoids. Thus

condition (c) for f is equivalent to condition (b) for f , which completes

the proof of the equivalence of the different parts of Definition 2.2.3.

We now proceed with the proof of Theorem 2.2.1. Note first of all

that the implications (1) ⇒ (2) and (2) ⇒ (3) are easy. We give a


short discussion of (1)⇒ (3) anyway, and then we prove (3)⇒ (2) and

(2)⇒ (1).

Note also that the equivalence (1)⇔ (2) is the content of Proposition

1.6 of [136]; we give a proof here because the proof of Proposition 1.6

was “left to the reader” in [136].

(1) ⇒ (3): Suppose A is a strict n-category satisfying condition (1).

This condition (from [136]) is compatible with truncation, so τ≤1(A)

satisfies condition (1) for 1-categories; which in turn is equivalent to

the standard condition of being a 1-groupoid, so we get that τ≤1(A)

is a 1-groupoid. On the other hand, the conditions (1) from [136] for

i-arrows, 1 ≤ i ≤ n, include the same conditions for the i − 1-arrows of

A(x, y) for any x, y ∈ Ob(A) (the reader has to verify this by looking at

the definition in [136]). Thus by the inductive statement of the present

theorem for strict n − 1-categories, A(x, y) is a strict n − 1-groupoid.

This shows that A satisfies condition (3).

(3) ⇒ (2): Suppose A is a strict n-category satisfying condition (3).

It already satisfies the first part of condition (2), by hypothesis. Thus

we have to show the second part, for example that for f : x → y in

Ob(A(x, y)), composition with f induces an equivalence

A(y, z) → A(x, z)

(the other part is dual and has the same proof which we won’t repeat

here).

In order to prove this, we need to make a digression about the effect

of composition with 2-morphisms. Suppose f, g ∈ Ob(A(x, y)) and sup-

pose that u is a 2-morphism from f to g—this last supposition may be

rewritten

u ∈ Ob(A(x, y)(f, g)).

Claim: Suppose z is another object; we claim that if composition with f

induces an equivalence A(y, z) → A(x, z), then composition with g also

induces an equivalence A(y, z) → A(x, z).

To prove the claim, suppose that h, k are two 1-morphisms from y to

z. We now obtain a diagram

HomA(y,z)(h, k) → HomA(x,z)(hf, kf)

HomA(x,z)(hg, kg)

↓

→ HomA(x,z)(hf, kg),

↓


where the top arrow is given by composition ∗0 with 1f ; the left arrow

by composition ∗0 with 1g; the bottom arrow by composition ∗1 with

the 2-morphism h ∗0 u; and the right morphism is given by composition

with k ∗0 u. This diagram commutes (that is the “Godement rule” or

“interchange rule” cf [136] p. 32). By the inductive statement of the

present theorem (version (2) of the groupoid condition) for the n − 1-

groupoid A(x, z), the morphisms on the bottom and on the right in the

above diagram are equivalences. The hypothesis in the claim that f is an

equivalence means that the morphism along the top of the diagram is an

equivalence; thus by the first part of Lemma 2.2.4 applied to the n− 2-

groupoids in the diagram, we get that the morphism on the left of the

diagram is an equivalence. This provides the second half of the criterion

(b) of Definition 2.2.3 for showing that the morphism of composition

with g, A(y, z) → A(x, z), is an equivalence of n− 1-groupoids.

To finish the proof of the claim, we now verify the first half of criterion

(b) for the morphism of composition with g (in this part we use directly

the condition (3) for A and don’t use either f or u). Note that τ≤1(A)

is a 1-groupoid, by the condition (3) which we are assuming. Note also

that (by definition)

π0A(y, z) = τ≤1A(y, z) and π0A(x, z) = τ≤1A(x, z),

and the morphism in question here is just the morphism of composition

by the image of g in τ≤1(A). Invertibility of this morphism in τ≤1(A)

implies that the composition morphism

(τ≤1A)(y, z) → (τ≤1A)(x, z)

is an isomorphism. This completes verification of the first half of criterion

(b), so we get that composition with g is an equivalence. This completes

the proof of the claim.

We now return to the proof of the composition condition for (2). The

fact that τ≤1(A) is a 1-groupoid implies that given f there is another

morphism h from y to x such that the class of fh is equal to the class of

1y in π0A(y, y), and the class of hf is equal to the class of 1x in π0A(x, x).

This means that there exist 2-morphisms u from 1y to fh, and v from

1x to hf . By the above claim (and the fact that the compositions with

1x and 1y act as the identity and in particular are equivalences), we get

that composition with fh is an equivalence

fh × A(y, z) → A(y, z),


and that composition with hf is an equivalence

hf × A(x, z) → A(x, z).

Let

ψf : A(y, z) → A(x, z)

be the morphism of composition with f , and let

ψh : A(x, z) → A(y, z)

be the morphism of composition with h. We have seen that ψhψf and

ψfψh are equivalences. By the second statement of Lemma 2.2.4 applied

to n− 1-groupoids, these imply that ψf is an equivalence.

The proof for composition in the other direction is the same; thus we

have obtained condition (2) for A.

(2) ⇒ (1): Look at the condition (1) by refering to [136]: in question

are the conditions GR′i,k and GR′′

i,k (i < k ≤ n) of Definition 1.1, p. 33

of [136]. By the inductive version of the present equivalence for n − 1-

groupoids and by the part of condition (2) which says that the A(x, y)

are n−1-groupoids, we obtain the conditions GR′i,k and GR′′

i,k for i ≥ 1.

Thus we may now restrict our attention to the condition GR′0,k and

GR′′0,k. For a 1-morphism a from x to y, the conditions GR′

0,k for all

k with respect to a, are the same as the condition that for all w, the

morphism of pre-multiplication by a

A(w, x) × a → A(w, y)

is an equivalence according to the version (c) of the notion of equivalence

(Definition 2.2.3). Thus, condition GR′0,k follows from the second part of

condition (2) (for pre-multiplication). Similarly condition GR′′0,k follows

from the second part of condition (2) for post-multiplication by every

1-morphism a. Thus condition (2) implies condition (1). This completes

the proof of Part (I) of the theorem.

For proof of the first part of Lemma 2.2.4, using the fact that iso-

morphisms of sets satisfy the same “three for two” property, and us-

ing the characterization of equivalences in terms of homotopy groups

(condition (a)) we immediately get two of the three statements: that

if f and g are equivalences then gf is an equivalence; and that if gf

and g are equivalences then f is an equivalence. Suppose now that gf

and f are equivalences; we would like to show that g is an equivalence.

First of all it is clear that if x ∈ Ob(A) then g induces an isomorphism


πi(B, f(x)) ∼= π0(C, gf(x)) (resp. π0(B) ∼= π0(C)). Suppose now that

y ∈ Ob(B), and choose a 1-morphism u going from y to f(x) for some

x ∈ Ob(A) (this is possible because f is surjective on π0). By condition

(2) for being a groupoid, composition with u induces equivalences along

the top row of the diagram

B(y, y) → B(y, f(x)) ← B(f(x), f(x))

C(g(y), g(y))

↓

→ C(g(y), gf(x))

↓

← C(gf(x), gf(x)).

↓

Similarly composition with g(u) induces equivalences along the bottom

row. The sub-lemma for n− 1-groupoids applied to the sequence

A(x, x) → B(f(x), f(x)) → C(gf(x), gf(x))

as well as the hypothesis that f is an equivalence, imply that the right-

most vertical arrow in the above diagram is an equivalence. Again ap-

plying the sub-lemma to these n− 1-groupoids yields that the leftmost

vertical arrow is an equivalence. In particular g induces isomorphisms

πi(B, y) = πi−1(B(y, y), 1y)∼=→ πi−1(C(g(y), g(y)), 1g(y)) = πi(C, g(y)).

This completes the verification of condition (a) for the morphism g,

completing the proof of part (IV) of the theorem.

Finally we prove the second part of Lemma 2.2.4 (from which we now

adopt the notations A,B, C,D, f, g, h). Note first of all that applying π0gives the same situation for maps of sets, so π0(g) is an isomorphism.

Next, suppose x ∈ Ob(A). Then we obtain a sequence

πi(A, x) → πi(B, f(x)) → πi(C, gf(x)) → πi(D, hgf(x)),

such that the composition of the first pair and also of the last pair are iso-

morphisms; thus g induces an isomorphism πi(B, f(x)) ∼= πi(C, gf(x)).

Now, by the same argument as for Part (IV) above, (using the hypoth-

esis that f induces a surjection π0(A) → π0(B)) we get that for any

object y ∈ Ob(B), g induces an isomorphism πi(B, y) ∼= πi(C, g(y)). By

Definition 2.2.3 (a) we have now shown that g is an equivalence. This

completes the proof of the statements in question.

Let nStrGpd be the category of strict n-groupoids. In Chapter 4 we

shall see that for any realization functor ℜ : nStrGpd → Top preserving

homotopy groups, topological spaces with nontrivial Whitehead prod-

ucts cannot be weakly equivalent to any ℜ(A). This was Grothendieck’s


motivation for proposing to look for a definition of weak n-category. Our

first look at the case of strict n-categories serves nevertheless as a guide

to the outlines of any general theory of weak n-categories.

3

Fundamental elements of n-categories

The observation that the theory of strict n-groupoids is not enough to

give a good model for homotopy n-types (detailed in the next Chapter 4),

led Grothendieck to ask for a theory of n-categories with weakly associa-

tive composition. This will be the main subject of our book, in particular

we use the terminology n-category to mean some kind of object in a pos-

sible theory with weak associativity, or even ill-defined composition, or

perhaps some other type of weakening (as will be briefly discussed in

Chapter 6).

There are a certain number of basic elements expected of any theory

of n-categories, and which can be explained without refering to a full

definition. It will be useful to start by considering these. Our discussion

follows Tamsamani’s paper [206], but really sums up the general expec-

tations for a theory of n-categories which were developped over many

years starting with Benabou and continuing through the theory of strict

n-categories and Grothendieck’s manuscript.

For this chapter, we will use the terminology “n-category” to mean

any object in a generic theory of n-categories. We will sometimes use the

idea that our generic theory should admit direct products and disjoint

sums.

3.1 A globular theory

We saw that a strict n-category has, in particular, an underlying globular

set. This basic structure should be conserved, in some form, in any weak

theory.

(OB)—An n-categoryA should have an underlying set of objects denoted


44 Fundamental elements of n-categories

Ob(A). If i = 0 then the structure A is identified with just this set

Ob(A), that is to say a 0-category is just a set.

(MOR)—If i ≥ 1 then for any two elements x, y ∈ Ob(A), there should

be an n−1-category of morphisms from x to y denoted MorA(x, y). From

these two things, we obtain by induction a whole family of sets called

the sets of i-morphisms of A for 0 ≤ i ≤ n.

(PS)—With respect to direct products and disjoint sums, we should have

Ob(A× B) = Ob(A)×Ob(B) and Ob(A ⊔ B) = Ob(A) ⊔Ob(B).

The set of i-morphisms of A can be defined inductively by the follow-

ing procedure. Put

Mor[A] :=∐

x,y∈Ob(A)

MorA(x, y);

this is the n− 1-category of morphisms of A.

By induction we obtain the n−i-category of i-morphisms ofA, denoted

by

Mori[A] := Mor[Mor[· · · [A] · · · ]].

This is defined whenever 0 ≤ i ≤ n, with Mor0[A] := A and Morn[A]

being a set.

Define

Mori[A] := Ob(Mori[A]).

This is a set, called the set of i-morphisms of A.

From the above definitions we can write

Mori[A] =∐

x,y∈Mori−1[A]

MorMori−1[A](x, y),

and by compatibility of objects with coproducts,

Mori[A] =∐

x,y∈Mori−1[A]

Ob(MorMori−1[A](x, y)).

In particular, we have maps si and ti from Mori[A] to Mori−1[A] taking

an element f ∈ Mori[A] lying in the piece of the coproduct indexed by

(x, y), to si(f) := x or ti(f) := y respectively. These maps are called

source and target and if no confusion arises, the index i may be dropped.

If u, v ∈ Mori[A], let Mori+1A (u, v) denote the preimage of the pair

(u, v) by the map (si+1, ti+1). It is nonempty only if si(u) = si(v) and

ti(u) = ti(v) and when using the notation Mori+1A (u, v) we generally

mean to say that these conditions are supposed to hold. Similarly, we

get n− i− 1-categories denoted Mori+1A (u, v).

3.1 A globular theory 45

In this way, starting just from the principles (OB) and (MOR) together

with the compatibility with sums in (PS), we obtain from an n-category

a collection of sets

Mor0[A] = Ob(A); Mor1[A], . . . ,Morn[A]

together with pairs of maps

si, ti : Mori[A] → Mori−1[A].

They satisfy

sisi+1 = siti+1, tisi+1 = titi+1.

These elements make our theory of n-categories into a globular theory.

Among other things, starting from this structure we can draw pictures

in a way which is usual for the theory of n-categories. These pictures

explain why the theory is called “globular”. A 0-morphism is just a

point, and a 1-morphism is pictured as a usual arrow

r r-

A 2-morphism is pictured as

r rR

⇓

whereas a 3-morphism should be thought of as a sort of “pillow” which

might be pictured as

s

s


3.2 Identities

For each x ∈ Ob(A) there should be a natural element 1x ∈ MorA(x, x),

called the identity of x. One can envision theories in which the identity is

not well-defined but exists only up to homotopy, see Kock and Joyal [140]

[128]. However, the theory considered here will have canonical identities.

Following the same inductive procedure as in the previous section, we

get morphisms for any 0 ≤ i < n,

ei : Mori[A] → Mori+1[A]

such that si+1ei(u) = u and ti+1ei(u) = u. We call ei(u) the identity

i+ 1-morphism of the i-morphism u.

Some authors introduce a category of globules Gn having objects Mi

for 0 ≤ i ≤ n, with generating morphisms si, ti : Mi → Mi−1 and

ei :Mi → Mi+1 subject to the relations

sisi+1 = siti+1, tisi+1 = titi+1, si+1ei = 1Mi , ti+1ei = 1Mi .

An n-globular set is a functor Gn → Set; with the identities this should

be called “reflexive”. Any n-category A induces an underlying globular

set constructed as above. Other authors (such as Batanin) use a category

of globules which doesn’t have the identity arrows ei, leading to non-

reflexive globular sets, indeed we shall use that notation in Section 6.3.

The first and basic idea for defining a theory of n-categories is that an

n-category should consist of an underlying globular set (with or without

identities), plus additional structural morphisms satisfying certain prop-

erties. Whereas the Batanin-type theories [20] [148] [162] are closest to

this ideal, the Segal-type theories we consider in the present book will

add additional structural sets to the basic globular set of A.

3.3 Composition, equivalence and truncation

For objects x, y, z ∈ Ob(A) there should be some kind of morphism of

n− 1-categories

MorA(x, y)×MorA(y, z) → MorA(x, z) (3.3.1)

corresponding to composition. In the Segal-type theories considered in

this book, the composition morphism is not well defined and may not

even exist, rather existing only in some homotopic sense.

Nonetheless, in order best to motivate the following discussion, assume

3.3 Composition, equivalence and truncation 47

for the moment that we know what composition means, particularly how

to define g f ∈ Mor1A(x, z) for f ∈Mor1A(x, y) and g ∈ Mor1A(x, y).

We can then inductively define a notion of equivalence. Tamsamani

calls this inner equivalence [206] to emphasize that we are speaking of

arrows in our n-category A which are equivalences internally in A. To

be more precise, we will define what it means for f ∈ Mor1A(x, y) to be

an inner equivalence between x and y. If such an f exists, we say that x

and y are equivalent and write x ∼ y.

Inductively we suppose known what this means for n − 1-categories,

and in particular within the n− 1-categories MorA(x, x) or MorA(y, y).

The definition then proceeds by saying that f ∈Mor1A(x, y) is an inner

equivalence between x and y, if there exists g ∈ Mor1A(y, x) such that

g f is equivalent to 1x in MorA(x, x) and f g is equivalent to 1y in

MorA(y, y).

This notion should be transitive in the sense that if f is an equivalence

from x to y and g is an equivalence from y to z, then g f should be an

equivalence from x to z. The relation “x ∼ y” is therefore a transitive

equivalence relation on the set Ob(A).

Define the truncation τ≤0(A) to be the quotient set Ob(A)/ ∼.

We can go further and define the 1-categorical truncation τ≤1(A), a

1-category, as follows:

Ob(τ≤1(A)) := Ob(A),

Mor1τ≤1(A)(x, y) := τ≤0(MorA(x, y)).

In other words, the objects of τ≤1(A) are the same as the objects of A,

but the morphisms of τ≤1(A) are the equivalence classes of 1-morphisms

of A, under the equivalence relation on the objects of the n− 1-category

MorA(x, y).

Composition of morphisms in τ≤1(A) should be defined by composing

representatives of the equivalence classes. One of the main requirements

for our theory of n-categories is that this composition in τ≤1(A) should

be well-defined, independent of the choice of representatives and indeed

independent of the choice of notion of composition morphism introduced

at the start of this section.

Denote also by ∼ the equivalence relation obtained in the same way on

the objects of the n− i-categories Mori[A]. Noting that it is compatible

with the source and target maps, we get an equivalence relation ∼ on

MoriA(u, v) for any i− 1-morphisms u and v.

The above discussion presupposed the existence of some kind of com-


position operation, but in the Segal-style theory we consider in this book,

such a composition morphism is not canonically defined. Thus, we restart

the discussion without assuming existence of a composition morphism

of n − 1-categories. The first fundamental structure to be considered is

thus:

(EQUIV)—on each set Mori[A] we have an equivalence relation ∼ com-

patible with the source and target maps, giving the set of i-morphisms

up to equivalence Mori[A]/ ∼. For i = n this equivalence relation should

be trivial. The induced relation on MoriA(u, v) is also denoted ∼.

We can then consider the structure of composition which is well-

defined up to equivalence, in other words it is given by a map on quotient

sets.

(COMP)—for any 0 < i ≤ n and any three i − 1-morphisms u, v, w

sharing the same sources and the same targets, we have a well-defined

composition map(MoriA(u, v)/ ∼

)×(MoriA(v, w)/ ∼

)→ MoriA(u,w)/ ∼

which is associative and has the classes of identity morphisms as left and

right units.

These two structures are compatible in the sense that composition is

defined after passing to the quotient by ∼. As a matter of simplifying

notation, given f ∈ MoriA(u, v) and g ∈ MoriA(v, w) then denote by

g f any representative in MoriA(u,w) for the composition of the class

of g with the class of f . This is well-defined up to equivalence and by

construction independent, up to equivalence, of the choices of represen-

tatives f and g for their equivalence classes.

Equivalence and composition also satisfy the following further com-

patibility condition, expressing the notion of equivalence in terms which

closely resemble the classical definition of equivalence of categories.

(EQC)—for any 0 ≤ i < n and u, v ∈MoriA sharing the same source and

target (i.e. si(u) = si(v) and ti(u) = ti(v) in case i > 0), then u ∼ v if

and only if there exist f ∈Mori+1A (u, v) and g ∈Mori+1

A (v, u) such that

g f ∼ 1u and f g ∼ 1v.

With these structures, we can define the 1-categories τ≤1MoriA(u, v),

having objects the elements of MoriA(u, v) and as morphisms between

w, z ∈ MoriA(u, v) the equivalence classes Mori+1A (w, z)/ ∼. The compo-

sition of (COMP) gives this a structure of 1-category, and w ∼ z if and

only if w and z are isomorphic objects of τ≤1MoriA(u, v). At the bottom

level we obtain a 1-category denoted τ≤1(A) and called the 1-truncation

of A, whose set of objects is Ob(A) and whose set of morphisms is


Mor1[A]/ ∼. These constructions are compatible with the induction in

the sense that τ≤1MoriA(u, v) is indeed the 1-truncation of the n − i-

category MoriA(u, v).

Suppose x, y ∈ Mori−1[A] and u, v ∈ MoriA(x, y). An element f ∈

Mori+1A (u, v) is said to be an internal equivalence between u and v, if its

class is an isomorphism in τ≤1MoriA(x, y). This is equivalent to requiring

the existence of g ∈Mori+1A (v, u) such that g f ∼ 1u and f g ∼ 1v.

3.4 Enriched categories

The natural first approach to the notion of n-category is to ask for n−1-

categories of morphisms MorA(x, y), with composition operations (3.3.1)

which are strictly associative and have the 1x as strict left and right

units. This gives a structure of category enriched over n− 1-categories.

In an intuitive sense the reader should think of an n-category in this

way. However, if the definition is applied inductively over n, that is to

say that the n − 1-categories MorA(x, y) are themselves enriched over

n− 2-categories and so forth, one gets to the notion of strict n-category

considered in the previous chapter. But, as we shall discuss in the next

Chapter 4 below, the strict n-categories are not sufficient to capture all

of the homotopical behavior we want for n ≥ 3.

Paoli has shown [170] that homotopy n-types can be modelled by

semistrict n-groupoids, in other words n-categories which are strictly

enriched over weak n − 1-categories. Bergner showed a corresponding

strictification theorem for Segal categories, and the analogous strictifi-

cation from A∞-categories to dg categories has been known to the ex-

perts for some time. Lurie’s technique [153] for constructing the model

category structure we consider here, gives additionally the strictification

theorem generalizing Bergner’s result. So, as we shall discuss briefly in

Section 21.5, the objects of our Segal-type theory of n-categories can

always be assumed equivalent to semistrict ones, that is to categories

strictly enriched over the model category for n − 1-precategories. This

doesn’t mean that we can go inductively towards strict n-categories be-

cause the strictification operation is not compatible with direct product,

so if applied to the enriched morphism objects it destroys the strict en-

richment structure. As Paoli notes in [170], semistrictification at one

level is as far as we can go.


3.5 The (n + 1)-category of n-categories

One of the main goals of a theory of n-categories is to provide a structure

of n + 1-category on the collection of all n-categories. Of course, some

discussion of universes is needed here: the collection of all n-categories in

a universeU should form an n+1-category in a bigger containing universe

V ⊃ U. This precision will be dropped from most of our discussions

below.

Recall that the notion of 2-category was originally introduced because

of the familiar observation that “the set of all categories is actually a

2-category”, a 2-category to be denoted 1CAT . Its objects are the 1-

categories (in the smaller universe); the 1-morphisms of 1CAT are the

functors between categories, and the 2-morphisms between functors are

the natural transformations.

In general, we hope and expect to obtain an n + 1-category denoted

nCAT , whose objects are the n-categories. The 1-morphisms of nCAT

are the “true” functors between n-categories, and we obtain all of the

Mori[nCAT ] for 0 ≤ i ≤ n + 1 which are higher analogues of natural

transformations and so on.

The notion of internal equivalence within nCAT itself, yields the no-

tion of external equivalence between n-categories: a functor f : A → B of

n-categories, by which we mean in the most general sense an element of

Mor1nCAT (A,B), is said to be an external equivalence if it is an internal

equivalence considered as a 1-morphism in nCAT .

In practice, a theory of n-categories will usually involve defining some

kind of mathematically structured set or collection of sets, which natu-

rally generates a usual 1-category of n-categories, which we can denote

by nCat. We expect then that Ob(nCat) = Ob(nCAT ) but that there

is a natural projection Mor1[nCat] → Mor1[nCAT ] compatible with

composition, indeed it should come from a morphism of n+1-categories

nCat → nCAT (which is to say, a 1-morphism in the n + 2-category

(n+ 1)CAT !).

However, the notion of external equivalence in nCat will not gen-

erally speaking have the same characterization as in nCAT : if f ∈

Mor1[nCat] projects to an equivalence in nCAT it means that there is

g ∈ Mor1[nCAT ] such that fg and gf are equivalent to identities; how-

ever the essential inverse g will not necessarily come from a morphism

in nCat. For precisely this reason, one of the main tasks needed to get a

theory of n-categories off the ground, is to give a different definition of

when a usual morphism f ∈ Mor1nCat(A,B) is an external equivalence.

3.5 The (n+ 1)-category of n-categories 51

Of course it is to be expected and—one hopes—later proven that this

standalone definition of external equivalence, should become equivalent

to the above definition once we have nCAT in hand.

A morphism of n-categories f : A → B should induce a morphism

of sets Ob(A) → Ob(B) (usually denoted just by x 7→ f(x)), and

for any x, y ∈ Ob(A) it should induce a morphism of n − 1-categories

MorA(x, y) → MorB(f(x), f(y)). Just as was the case for composition,

the morphism part of f needn’t necessarily be very well defined, but it

should be well-defined up to an appropriate kind of equivalence.

It is now possible to state, by induction on n, the second or “stan-

dalone” definition of external equivalence. A morphism f : A → B

is said to be fully faithful if for every x, y ∈ Ob(A) the morphism

MorA(x, y) → MorB(f(x), f(y)) is an external equivalence between n−

1-categories. And f is said to be essentially surjective if it induces a sur-

jection Ob(A) ։ Ob(B)/ ∼. Then f is said to be an external equivalence

if it is fully faithful and essentially surjective.

We can state the required compatibility between the two notions:

(EXEQ)—a morphism f : A → B, an element of Mor1nCAT (A,B), is

an external equivalence (fully faithful and essentially surjective), if and

only if it is an inner equivalence in nCAT (i.e. has an essential inverse

g such that fg and gf are equivalent to the identities).

Note that the fully faithful condition implies (by an inductive consid-

eration and comparison with the truncation operation) that if f : A → B

is an equivalence in either of the two equivalent senses, and if a, b are

i-morphisms of A with the same source and target, then the set of inner

equivalence classes of i + 1-morphisms from a to b in A, is isomorphic

via f to the set of inner equivalence classes of i+1-morphisms from f(a)

to f(b) in B.

When developing a theory of n-categories, we therefore expect to

be in the following situation: having first obtained a 1-category of n-

categorical structures denoted nCat, we obtain a notion of when a mor-

phism f in this category, or first kind of functor, is an external equiv-

alence. On the other hand, in the full n + 1-category f should have an

essential inverse g. So, one of the main steps towards construction of

nCAT is to formally invert the external equivalences. This is a typical

localization problem. Furthermore nCAT should be closed under limits

and colimits, so it is very natural to use Quillen’s theory of model cat-

egories, and all of the localization machinery that is now known to go

along with it, as our main technical tool for going towards the construc-

tion of nCAT .


To finish this section, note one of the interesting and important fea-

tures of nCAT : it is, in a certain sense, enriched over itself. In other

words, for two objects A,B ∈ Ob(nCAT ), we get an n-category of mor-

phisms MornCAT (A,B) which, since it is an n-category (and furthermore

in the same universe level as A and B), is itself an object of nCAT :

MornCAT (A,B) ∈ Ob(nCAT ).

This is the motivation for using the theory of cartesian model categories

with internal Hom as a preliminary model for nCAT .

3.6 Poincare n-groupoids

An n-category is said to be an n-groupoid if all i-morphisms are in-

ner equivalences. More generally, we say that A is k-groupic if all i-

morphisms are inner equivalences for i > k. Lurie introduces the nota-

tion (n, k)-category for a k-groupic n-category, mostly used in the limit-

ing case n =∞.

Fundamental to Grothendieck’s vision in “Pursuing stacks” was the

Poincare n-groupoid of a space. If X is a topological space, this is to be

an n-groupoid denoted by Πn(X), with the following properties:

Ob(Πn(X)) = X,

for 0 ≤ i < n

Mori[Πn(X)] = C0glob([0, 1]

i, X)

where the right hand side is the subset of maps of the i-cube into X

satisfying certain globularity conditions (explained below), and at i = n

we have

Morn[Πn(X)] = C0glob([0, 1]

n, X)/ ∼

where ∼ is an equivalence relation similar to the one considered above

(and indeed, it is the same in the context of Πk(X) for k > n).

The globularity condition is automatic for i = 1, thus the 1-morphisms

in Πn(X) are continuous paths p : [0, 1] → X with source s1(p) := p(0)

and target t1(p) := p(1). In the limiting case n = 1, the 1-morphisms in

Π1(X) are homotopy classes of paths with homotopies fixing the source

and target, and Π1(X) is just the classical Poincare groupoid of the

space X .

The globularity condition is most easily understood in the case i = 2:

3.7 Interiors 53

a 2-morphism in Πn(X) should be a homotopy between paths, that is

to say it should be a map ψ : [0, 1]2 → X such that ψ(0, t) and ψ(1, t)

are independent of t.

For 2 ≤ i ≤ n, the globularity condition on a map ψ : [0, 1]i → X

says that for any 0 ≤ k < i and any z1, . . . , zk−1 ∈ [0, 1], the functions

(zk+1, . . . , zi) 7→ ψ(z1, . . . , zk−1, 0, zk+1, . . . , zi)

and

(zk+1, . . . , zi) 7→ ψ(z1, . . . , zk−1, 1, zk+1, . . . , zi)

are constant in (zk+1, . . . , zn). The source and target of ψ are defined by

siψ : (z1, . . . , zi−1) 7→ ψ(z1, . . . , zi−1, 0),

tiψ : (z1, . . . , zi−1) 7→ ψ(z1, . . . , zi−1, 1).

Grothendieck’s fundamental prediction was that this globular set should

have a natural structure of n-groupoid denoted Πn(X); that there should

be a realization construction taking an n-groupoid G to a topological

space |G|; and that these two constructions should set up an equiva-

lence of homotopy theories between n-truncated spaces (i.e. spaces with

πi(X) = 0 for i > n) and n-groupoids. This would generalize the clas-

sical correspondence between 1-groupoids and their classifying spaces

which are disjoint unions of Eilenberg-MacLane K(π, 1)-spaces.

3.7 Interiors

A useful notion which should exist in a theory of n-categories is the

notion of k-groupic interior denoted Intk(A). This is the largest sub-n-

category of A which is k-groupic, and we should have

Mori[Intk(A)] = Mori[A], i ≤ k

whereas for i > k the i-morphisms of the interior Mori[Intk(A)] should

consist only of those i-morphisms of A which are equivalences, i.e. in-

vertible up to equivalence. Specifying exactly the structure of Intk(A)

will depend on the particular theory of n-categories, but in any case it

should be a k-groupic n-category i.e. an (n, k)-category.

The usual case is for k = 1, and sometimes the index 1 will then

be forgotten. Thus, if A is an n-category then we get a 1-groupic n-

category Int(A). As will be discussed below in the next section and


more extensively in Chapter 5, the notion of 1-groupic n-category is

well modeled by the notions of simplicial category, Segal category, Rezk

complete Segal space, or quasicategory. Simplicial categories typically

arise as Dwyer-Kan localizations L( ) of model categories, and one

feature of the model category PCn(Set) constructed in this book is

that

Int(nCAT ) = L(PCn(Set)).

This adds to the motivation for why it is interesting and important to

use model categories as a substrate for the theory of n-categories: we get

a calculatory model for an important piece of nCAT namely its interior.

3.8 The case n =∞

Constructing a theory of∞-categories in general, represents a new level

of difficulty and to do this in detail would go beyond the scope of the

present book. We include here a few comments about this problem,

largely following Cheng’s observation [64]. The example considered in

her paper shows that in an algebraic sense, any ∞-category whose i-

morphisms have duals at all levels, looks like an∞-groupoid. However, it

is clear that we don’t want to identify such∞-categories with duals, and

∞-groupoids. Indeed, they occupy almost dual positions in the general

theory as predicted by Baez and Dolan. From this paradox one can

conclude that the notion of “equivalence” in an∞-category is not merely

an algebraic one. One way around this problem would be to include the

notion of equivalence in the initial structure of an ∞-category: it would

be a globular set with additional structure similar to the structure used

for the case of n-categories; but also with the information of a subset

of the set of i-morphisms which are to be designated as “equivalences”.

These subsets would be required to satisfy a compatibility condition

similar to (EXEQ) above. Then, in Cheng’s example [64] there would be

two compatible choices: either to designate everybody as an equivalence,

in which case we get an ∞-groupoid; or to designate only some (or

potentially none other than the identities) as equivalences, yielding an

“∞-category with duals” more like what Baez and Dolan are looking

for.

In view of these problems, it is tempting to take a shortcut towards

consideration of certain types of ∞-categories. The shortcut is moti-

vated by Grothendieck’s Poincare n-groupoid correspondence, which he

3.8 The case n =∞ 55

says should also extend to an equivalence between∞-groupoids and the

homotopy theory of all CW-complexes. Turning this on its head, we

can use that idea to define the notion of ∞-groupoid as simply being

a homotopy type of a space. The iterative enrichment procedure yields

the notion of n-categories when started from 0-categories being sets. If

instead we start with ∞-groupoids being spaces, then iterating gives a

notion of (∞, n)-categories which are n-groupic ∞-categories, i.e. ones

whose i-morphisms are supposed and declared to be invertible for all

i > n.

At the first stage, an (∞, 1)-category is therefore a category enriched

over spaces. In Part II we will consider in detail many of the different

current approaches to this theory. At the n-th stage, in the Segal-type

theory pursued here, we obtain the notion of Segal n-category. The possi-

bility of doing an iterative definition in various different cases, motivates

our presentation here of a general iterative construction of the theory of

categories weakly enriched over a cartesian model category. The carte-

sian condition corresponds to the fact that the composition morphism

goes out from a product, so the model structure should have a good com-

patibility property with respect to direct products. When the iteration

starts from the model category of sets, we get the theory of n-categories;

and starting from the model category of spaces leads to the theory of

Segal n-categories which is one approach to (∞, n)-categories.

Looking only at (∞, n)-categories rather than all∞-categories is com-

patible with the notion of nCAT , in the sense that (∞, n)CAT , the col-

lection of all (∞, n)-categories, is expected to have a natural structure of

(∞, n+ 1)-category. In the theory presented here, this will be achieved

by constructing a cartesian model category for Segal n-categories. The

cartesian property thus shows up at the output side of the iteration,

and at the input side because we need to handle products in order to

talk about weak composition morphisms. Thus, one of our main goals is

to construct an iteration step starting with a cartesian model category

M and yielding a cartesian model category PC(M ) representing the

homotopy theory of weakly M -enriched categories.

4

The need for weak composition

In this chapter, we take a break from the general theory to consider

the problem of realization of homotopy 3-types. It is here that the phe-

nomenon of “weak composition” shows up first, in that strict 3-groupoids

are not sufficient to model all 3-truncated homotopy types. The nota-

tions here refer and continue those of Chapter 2.

The classical Eckmann-Hilton argument, originally used to show that

the homotopy groups πi are abelian for i ≥ 2, applies in the context of

strict n-categories to give a vanishing of certain homotopy operations.

Indeed, not only the πi but the i-th loop spaces are abelian objects, and

this forces the Whitehead products to vanish. This observation, which I

learned from G. Maltsiniotis and A. Bruguieres, had been used by many

people to argue that strict n-categories do not contain sufficient infor-

mation to model homotopy n-types, as soon as n ≥ 3. See for example

Brown [56], with Gilbert [57] and with Higgins [58] [59]; Grothendieck’s

discussion of this in various places in [108], and the paper of Berger [29].

In R. Brown’s terminology, strict n-groupoids correspond to crossed

complexes. While a nontrivial action of π1 on the πi can occur in a

crossed complex, the higher Whitehead operations such as π2⊗π2 → π3must vanish. The Eckmann-Hilton argument for strict n-categories is

also known as the “interchange rule” or “Godement relation”. This effect

occurs when one takes two 2-morphisms a and b both with source and

target a 1-identity 1x. There are various ways of composing a and b

in this situation, and comparison of these compositions leads to the

conclusion that all of the compositions are commutative. In a weak n-

category, this commutativity would only hold up to higher homotopy,

which leads to the notion of “braiding”; and in fact it is exactly the

braiding which leads to the Whitehead operation. However, in a strict


4.1 Realization functors 57

n-groupoid, the commutativity is strict and applies to all higher arrows,

so the Whitehead operation is trivial.

The same may be said in the setting of 3-categories not necessarily

groupoids: there are some examples (which G. Maltsiniotis pointed out

to me) in Gordon-Power-Street [104] of weak 3-categories not equivalent

to strict ones. This in turn is related to the difference between braided

monoidal categories and symmetric monoidal categories, see for example

the nice discussion in Baez-Dolan [6].

This chapter gives a variant on these observations; it is a modified

version of the preprint [197]. We will show that one cannot obtain all

homotopy 3-types by any reasonable realization functor from strict 3-

groupoids (i.e. groupoids in the sense of [136]) to spaces. More precisely

we show that one does not obtain the 3-type of S2. This constitutes a

small generalization of Berger’s theorem [29], which concerned the stan-

dard realization functor. We define the notion of possible “reasonable

realization functor” in Definition 4.1.1 to be any functor ℜ from the

category of strict n-groupoids to Top, provided with a natural transfor-

mation r from the set of objects of G to the points of ℜ(G), and natural

isomorphisms π0(G) ∼= π0(ℜ(G)) and πi(G, x) ∼= πi(ℜ(G), r(x)). This ax-

iom is fundamental to the question of whether one can realize homotopy

types by strict n-groupoids, because one wants to read off the homotopy

groups of the space from the strict n-groupoid. The standard realization

functors satisfy other properties beyond this minimal one.

In order to apply Definition 4.1.1, the interchange argument is written

in a particular way. We get a picture of strict 3-groupoids having only one

object and one 1-morphism, as being equivalent to abelian monoidal ob-

jects (G,+) in the category of groupoids, such that (π0(G),+) is a group.

In the case in question, this group will be π2(S2) = Z. Then comes the

main part of the argument. We show that, up to inverting a few equiva-

lences, such an object has a morphism giving a splitting of the Postnikov

tower (Proposition 4.4.1). It follows that for any realization functor re-

specting homotopy groups, the Postnikov tower of the realization (which

has two stages corresponding to π2 and π3) splits. This implies that the

3-type of S2 cannot occur as a realization, Theorem 4.4.2.

4.1 Realization functors

Recall that nStrGpd is the category of strict n-groupoids as defined

in Chapter 2. Let Top be the category of topological spaces. The fol-

58 The need for weak composition

lowing definition encodes the minimum of what one would expect for a

reasonable realization functor from strict n-groupoids to spaces.

Definition 4.1.1 A realization functor for strict n-groupoids is a func-

tor

ℜ : nStrGpd → Top

together with the following natural transformations:

r : Ob(A) → ℜ(A);

ζi(A, x) : πi(A, x) → πi(ℜ(A), r(x)),

the latter including ζ0(A) : π0(A) → π0(ℜ(A)); such that the ζi(A, x)

and ζ0(A) are isomorphisms for 0 ≤ i ≤ n, such that ζ0 takes the iso-

morphism class of x to the connected component of r(x), and such that

the πi(ℜ(A), y) vanish for i > n.

Theorem 4.1.2 There exists a realization functor ℜ for strict n-

groupoids.

Kapranov and Voevodsky [136] construct such a functor. Their con-

struction proceeds by first defining a notion of “diagrammatic set”; they

define a realization functor from n-groupoids to diagrammatic sets (de-

noted Nerv), and then define the topological realization of a diagram-

matic set (denoted |·|). The composition of these two constructions gives

a realization functor

G 7→ ℜKV (G) := |Nerv(G)|

from strict n-groupoids to spaces. Note that this functor ℜKV satisfies

the axioms of 4.1.1 as a consequence of Propositions 2.7 and 3.5 of [136].

One obtains a different construction by considering strict n-groupoids

as weak n-groupoids in the sense of [206] (multisimplicial sets) and then

taking the realization of [206]. This construction is actually due to the

Australian school many years beforehand—see [29]—and we call it the

standard realization ℜstd. The properties of 4.1.1 can be extracted from

[206] (although again they are classical results).

We don’t claim here that any two realization functors must be the

same. This is why we shall work, in what follows, with an arbitrary

realization functor satisfying the axioms of 4.1.1.

Proposition 4.1.3 If C → C′ is a morphism of strict n-groupoids in-

ducing isomorphisms on the πi then ℜ(C) → ℜ(C′) is a weak homotopy

4.2 n-groupoids with one object 59

equivalence. Conversely if f : C → C′ is a morphism of strict n-groupoids

which induces a weak equivalence of realizations then f was an equiva-

lence.

Proof Apply version (a) of the equivalent conditions in Definition 2.2.3,

together with the property of Definition 4.1.1.

4.2 n-groupoids with one object

Let C be a strict n-category with only one object x. Then C is an n-

groupoid if and only if C(x, x) is an n−1-groupoid and π0C(x, x) (which

has a structure of monoid) is a group. This is version (3) of the definition

of groupoid in Theorem 2.2.1. Iterating this remark one more time we

get the following statement.

Lemma 4.2.1 The construction of 2.1.2 establishes an equivalence of

categories between the strict n-groupoids having only one object and only

one 1-morphism, and the abelian monoid-objects G in (n−2)StrGpd such

that the monoid π0(G) is a group.

Proof Lemma 2.1.2 gives an equivalence between the categories of abelian

monoid-objects in (n − 2)StrCat, and the strict n-categories having

only one object and one 1-morphism. The groupoid condition for the

n-category is equivalent to saying that G is a groupoid, and that π0(G)

is a group.

Corollary 4.2.2 Suppose C is a strict n-category having only one ob-

ject and only one 1-morphism, and let B be the strict n+ 1-category of

2.1.3 with one object b and B(b, b) = C. Then B is a strict n+1-groupoid

if and only if C is a strict n-groupoid.

Proof: Keep the notation U of the proof of 2.1.3. If C is a groupoid

this means that G satisfies the condition that π0(G) be a group, which in

turn implies that U is a groupoid. Note that π0(U) = ∗ is automatically

a group; so applying the observation 4.2.1 once again, we get that B is

a groupoid. In the other direction, if B is a groupoid then C = B(b, b) is

a groupoid by versions (2) and (3) of the definition of groupoid.


4.3 The case of the standard realization

Before getting to our main result which concerns an arbitrary realiza-

tion functor satisfying 4.1.1, we take note of an easier argument which

shows that the standard realization functor cannot give rise to arbitrary

homotopy types.

Definition 4.3.1 A collection of realization functors ℜn for n-groupoids

(0 ≤ n < ∞) satisfying 4.1.1 is said to be compatible with looping if

there exist transformations natural in an n-groupoid A and an object

x ∈ Ob(A),

ϕ(A, x) : ℜn−1(A(x, x)) → Ωr(x)ℜn(A)

(where Ωr(x) means the space of loops based at r(x)), such that for i ≥ 1

the following diagram commutes:

πi(A, x)= πi−1(A(x, x), 1x) → πi−1(ℜn−1(A(x, x)), r(1x))

πi(ℜn(A), r(x))

↓

← πi−1(Ωr(x)ℜn(A), cst(r(x)))

↓

where the top arrow is ζi−1(A(x, x), 1x), the left arrow is ζi(A, x), the

right arrow is induced by ϕ(A, x), and the bottom arrow is the canonical

arrow from topology. (When i = 1, suppress the basepoints in the πi−1

in the diagram.)

Remark: The arrows on the top, the bottom and the left are isomor-

phisms in the above diagram, so the arrow on the right is an isomorphism

and we obtain as a corollary of the definition that the ϕ(A, x) are actu-

ally weak equivalences.

Remark: The collection of standard realizations ℜnstd for n-groupoids,

is compatible with looping. We leave this as an exercise.

Recall the statements of 2.1.3 and 4.2.2: if A is a strict n-category

with only one object x and only one 1-morphism 1x, then there exists a

strict n+ 1-category B with one object y, and with B(y, y) = A; and A

is a strict n-groupoid if and only if B is a strict n+ 1-groupoid.

Corollary 4.3.2 Suppose ℜn is a collection of realization functors

4.1.1 compatible with looping 4.3.1. Then if A is a 1-connected strict

n-groupoid (i.e. π0(A) = ∗ and π1(A, x) = 1), the space ℜn(A) is

weak-equivalent to a loop space.

4.4 Nonexistence of strict 3-groupoids giving rise to the 3-type of S261

Proof: Let A′ ⊂ A be the sub-n-category having one object x and one

1-morphism 1x. For i ≥ 2 the inclusion induces isomorphisms

πi(A′, x) ∼= πi(A, x),

and in view of the 1-connectedness of A this means (according to the

Definition 2.2.3 (a)) that the morphism A′ → A is an equivalence. It

follows (by definition 4.1.1) that ℜn(A′) → ℜn(A) is a weak equivalence.

Now A′ satisfies the hypothesis of 2.1.3, 4.2.2 as recalled above, so there

is an n+ 1-groupoid B having one object y such that A′ = B(y, y). By

the definition of “compatible with looping” and the subsequent remark

that the morphism ϕ(B, y) is a weak equivalence, we get that ϕ(B, y)

induces a weak equivalence

ℜn(A′) → Ωr(y)ℜn+1(B).

Thus ℜn(A) is weak-equivalent to the loop-space of ℜn+1(B).

The following corollary is due to C. Berger [29] (although the same

statement appears without proof in Grothendieck [108]). See also R.

Brown and coauthors [56] [57] [58] [59].

Corollary 4.3.3 (C. Berger [29]) There is no strict 3-groupoid A such

that the standard realization ℜstd(A) is weak-equivalent to the 3-type of

S2.

Proof: The 3-type of S2 is not a loop-space. By the previous corollary

(and the fact that the standard realizations are compatible with looping,

which we have above left as an exercise for the reader), it is impossible

for ℜstd(A) to be the 3-type of S2.

4.4 Nonexistence of strict 3-groupoids giving rise to

the 3-type of S2

The present discussion aims to extend Berger’s negative result to any

realization functor satisfying the minimal definition 4.1.1.

The first step is to prove the following statement (which contains the

main part of the argument). It basically says that the Postnikov tower

of a simply connected strict 3-groupoid C, splits. The intermediate B is

not really necessary for the statement but corresponds to the technique

of proof.

Proposition 4.4.1 Suppose C is a strict 3-groupoid with an object c


such that π0(C) = ∗, π1(C, c) = 1, π2(C, c) = Z and π3(C, c) = H for

an abelian group H. Then there exists a diagram of strict 3-groupoids

C ←gB ←

fA

h→ D

with objects b ∈ Ob(B), a ∈ Ob(A), d ∈ Ob(D) such that f(a) = b,

g(b) = c, h(a) = d. The diagram is such that g and f are equivalences of

strict 3-groupoids, and such that π0(D) = ∗, π1(D, d) = 1, π2(D, d) =

0, and such that h induces an isomorphism

π3(h) : π3(A, a) = H∼=→ π3(D, d).

Proof Start with a strict groupoid C and object c, satisfying the hy-

potheses of 4.4.1.

The first step is to construct (B, b). We let B ⊂ C be the sub-3-category

having only one object b = c, and only one 1-morphism 1b = 1c. We set

HomB(b,b)(1b, 1b) := HomC(c,c)(1c, 1c),

with the same composition law. The map g : B → C is the inclusion.

Note first of all that B is a strict 3-groupoid. This is easily seen using

version (1) of the definition in Theorem 2.2.1 (but one has to look at the

conditions in [136]). We can also verify it using condition (3). Of course

τ≤1(B) is the 1-category with only one object and only one morphism,

so it is a groupoid. We have to verify that B(b, b) is a strict 2-groupoid.

For this, we again apply condition (3) of 2.2.1. Here we note that

B(b, b) ⊂ C(c, c)

is the full sub-2-category with only one object 1b = 1c. Therefore, in

view of the definition of τ≤1, we have that

τ≤1(B(b, b)) ⊂ τ≤1(C(c, c))

is a full subcategory. A full subcategory of a 1-groupoid is again a 1-

groupoid, so τ≤1(B(b, b)) is a 1-groupoid. Finally, HomB(b,b)(1b, 1b) is

a 1-groupoid since by construction it is the same as HomC(c,c)(1c, 1c)

(which is a groupoid by condition (3) applied to the strict 2-groupoid

C(c, c)). This shows that B(b, b) is a strict 2-groupoid an hence that B is

a strict 3-groupoid.

Next, note that π0(B) = ∗ and π1(B, b) = 1. On the other hand, for

i = 2, 3 we have

πi(B, b) = πi−2(HomB(b,b)(1b, 1b), 12b)


and similarly

πi(C, c) = πi−2(HomC(c,c)(1c, 1c), 12c),

so the inclusion g induces an equality πi(B, b)=→ πi(C, c). Therefore,

by definition (a) of equivalence 2.2.3, g is an equivalence of strict 3-

groupoids. This completes the construction and verification for B and

g.

Before getting to the construction of A and f , we analyze the strict

3-groupoid B in terms of the discussion of 2.1.2 and 4.2.1. Let

G := HomB(b,b)(1b, 1b).

It is an abelian monoid-object in the category of 1-groupoids, with

abelian operation denoted by + : G × G → G and unit element de-

noted 0 ∈ G which is the same as 1b. The operation + corresponds to

both of the compositions ∗0 and ∗1 in B.

The hypotheses on the homotopy groups of C also hold for B (since g

was an equivalence). These translate to the statements that (π0(G),+) =

Z and G(0, 0) = H .

We now construct A and f via 2.1.2 and 4.2.1, by constructing a

morphism (G′,+) → (G,+) of abelian monoid-objects in the category

of 1-groupoids. We do this by a type of “base-change” on the monoid of

objects, i.e. we will first define a morphism Ob(G′) → Ob(G) and then

define G′ to be the groupoid with object set Ob(G′) but with morphisms

corresponding to those of G.

To accomplish the “base-change”, start with the following construc-

tion. If S is a set, let codsc(S) denote the groupoid with S as set of

objects, and with exactly one morphism between each pair of objects. If

S has an abelian monoid structure then codsc(S) is an abelian monoid

object in the category of groupoids.

Note that for any groupoid U there is a morphism of groupoids

U −→ codsc(Ob(U)),

and by “base change” we mean the following operation: take a set S

with a map p : S → Ob(U) and look at

V := codsc(S)×codsc(Ob(U)) U .

This is a groupoid with S as set of objects, and with

V(s, t) = U(p(s), p(t)).


A similar construction will be used later in Chapter 12 under the nota-

tion V = p∗(U). For the present purposes, note that if U is an abelian

monoid object in the category of groupoids, if S is an abelian monoid

and if p is a map of monoids then V is again an abelian monoid object

in the category of groupoids.

Apply this as follows. Starting with (G,+) corresponding to B via 2.1.2

and 4.2.1 as above, choose objects a, b ∈ Ob(G) such that the image of

a in π0(G) ∼= Z corresponds to 1 ∈ Z, and such that the image of b in

π0(G) corresponds to −1 ∈ Z. Let N denote the abelian monoid, product

of two copies of the natural numbers, with objects denoted (m,n) for

nonnegative integers m,n. Define a map of abelian monoids

p : N → Ob(G)

by

p(m,n) := m · a+ n · b := a+ a+ . . .+ a + b+ b+ . . .+ b.

Note that this induces the surjection N → π0(G) = Z given by (m,n) 7→

m− n.

Define (G′,+) as the base-change

G′ := codsc(N)×codsc(Ob(G)) G,

with its induced abelian monoid operation +. We have

Ob(G′) = N,

and the second projection p2 : G′ → G (which induces p on object sets)

is fully faithful i.e.

G′((m,n), (m′, n′)) = G(p(m,n), p(m′, n′)).

Note that π0(G′) = Z via the map induced by p or equivalently p2. To

prove this, say that: (i) N surjects onto Z so the map induced by p

is surjective; and (ii) the fact that p2 is fully faithful implies that the

induced map π0(G′) → π0(G) = Z is injective.

We let A be the strict 3-groupoid corresponding to (G′,+) via 2.1.2,

and let f : A → B be the map corresponding to p2 : G′ → G again via

2.1.2. Let a be the unique object of A (it is mapped by f to the unique

object b ∈ Ob(B)).

The fact that (π0(G′),+) = Z is a group implies that A is a strict

3-groupoid (4.2.1). We have π0(A) = ∗ and π1(A, a) = 1. Also,

π2(A, a) = (π0(G′),+) = Z


and f induces an isomorphism from here to π2(B, b) = (π0(G),+) = Z.

Finally (using the notation (0, 0) for the unit object of (N,+) and the

notation 0 for the unit object of Ob(G)),

π3(A, a) = G′((0, 0), (0, 0)),

and similarly

π3(B, b) = G(0, 0) = H ;

the map π3(f) : π3(A, a) → π3(B, b) is an isomorphism because it is the

same as the map

G′((0, 0), (0, 0)) → G(0, 0)

induced by p2 : G′ → G, and p2 is fully faithful. We have now completed

the verification that f induces isomorphisms on the homotopy groups, so

by version (a) of the definition of equivalence 2.2.3, f is an equivalence

of strict 3-groupoids.

We now construct D and define the map h by an explicit calculation

in (G′,+). First of all, let [H ] denote the 1-groupoid with one object

denoted 0, and with H as group of endomorphisms:

[H ](0, 0) := H.

This has a structure of abelian monoid-object in the category of groupoids,

denoted ([H ],+), because H is an abelian group. Let D be the strict 3-

groupoid corresponding to ([H ],+) via 2.1.2 and 4.2.1. We will construct

a morphism h : A → D via 2.1.2 by constructing a morphism of abelian

monoid objects in the category of groupoids,

h : (G′,+) → ([H ],+).

We will construct this morphism so that it induces the identity morphism

G′((0, 0), (0, 0)) = H → [H ](0, 0) = H.

This will insure that the morphism h has the property required for 4.4.1.

The object (1, 1) ∈ N goes to 0 ∈ π0(G′) ∼= Z. Thus we may choose

an isomorphism ϕ : (0, 0) ∼= (1, 1) in G′. For any k let kϕ denote the

isomorphism ϕ + . . . + ϕ (k times) going from (0, 0) to (k, k). On the

other hand, H is the automorphism group of (0, 0) in G′. The operations

+ and composition coincide on H . Finally, for any (m,n) ∈ N let 1m,ndenote the identity automorphism of the object (m,n). Then any arrow

α in G may be uniquely written in the form

α = 1m,n + kϕ+ u


with (m,n) the source of α, the target being (m+ k, n+ k), and where

u ∈ H .

We have the following formulae for the composition of arrows in G′.

They all come from the basic rule

(α β) + (α′ β′) = (α + α′) (β + β′)

which in turn comes simply from the fact that + is a morphism of

groupoids G′ × G′ → G′ defined on the cartesian product of two copies

of G. Note in a similar vein that 10,0 acts as the identity for the operation

+ on arrows, and also that

1m,n + 1m′,n′ = 1m+m′,n+n′ .

Our first equation is

(1l,l + kϕ) lϕ = (k + l)ϕ.

To prove this note that lϕ+ 10,0 = lϕ and our basic formula says

(1l,l lϕ) + (kϕ 10,0) = (1l,l + kϕ) (lϕ+ 10,0)

but the left side is just lϕ+ kϕ = (k + l)ϕ.

Now our basic formula, for a composition starting with (m,n), going

first to (m+ l, n+ l), then going to (m+ l+ k, n+ l+ k), gives

(1m+l,n+l + kϕ+ u) (1m,n + lϕ+ v)

= (1m,n + 1l,l + kϕ+ u) (1m,n + lϕ+ v)

= 1m,n 1m,n + (1l,l + kϕ) lϕ+ u v

= 1m,n + (k + l)ϕ+ (u v)

where of course u v = u+ v.

This formula shows that the morphism h from arrows of G′ to the

group H , defined by

h(1m,n + kϕ+ u) := u

is compatible with composition. This implies that it provides a morphism

of groupoids h : G → [H ] (recall from above that [H ] is defined to be the

groupoid with one object whose automorphism group isH). Furthermore

the morphism h is obviously compatible with the operation + since

(1m,n + kϕ+ u) + (1m′,n′ + k′ϕ+ u′) =


(1m+m′,n+n′ + (k + k′)ϕ+ (u+ u′))

and once again u + u′ = u u′ (the operation + on [H ] being given by

the commutative operation on H).

This completes the construction of a morphism h : (G,+) → ([H ],+)

which induces the identity on Hom(0, 0). This corresponds to a mor-

phism of strict 3-groupoids h : A → D as required to complete the

proof of Proposition 4.4.1.

We can now give the nonrealization statement.

Theorem 4.4.2 Let ℜ be any realization functor satisfying the prop-

erties of Definition 4.1.1. Then there does not exist a strict 3-groupoid

C such that ℜ(C) is weak-equivalent to the 3-truncation of the homotopy

type of S2.

Proof Suppose for the moment that we know Proposition 4.4.1; with

this we will prove 4.4.2. Fix a realization functor ℜ for strict 3-groupoids

satisfying the axioms 4.1.1, and assume that C is a strict 3-groupoid such

that ℜ(C) is weak homotopy-equivalent to the 3-type of S2. We shall

derive a contradiction.

Apply Proposition 4.4.1 to C. Choose an object c ∈ Ob(C). Note that,

because of the isomorphisms between homotopy sets or groups 4.1.1, we

have π0(C) = ∗, π1(C, c) = 1, π2(C, c) = Z and π3(C, c) = Z, so 4.4.1

applies with H = Z. We obtain a sequence of strict 3-groupoids

C ←gB ←

fA

h→ D.

This gives the diagram of spaces

ℜ(C) ←ℜ(g)

ℜ(B) ←ℜ(f)

ℜ(A)ℜ(h)→ ℜ(D).

The axioms 4.1.1 for ℜ imply that ℜ transforms equivalences of strict

3-groupoids into weak homotopy equivalences of spaces. Thus ℜ(f) and

ℜ(g) are weak homotopy equivalences and we get that ℜ(A) is weak

homotopy equivalent to the 3-type of S2.

On the other hand, again by the axioms 4.1.1, we have that ℜ(D) is

2-connected, and π3(ℜ(D), r(d)) = H (via the isomorphism π3(D, d) ∼=H induced by h, f and g). By the Hurewicz theorem there is a class

η ∈ H3(ℜ(D), H) which induces an isomorphism

Hur(η) : π3(ℜ(D), r(d))∼=→ H.

Here

Hur : H3(X,H) → Hom(π3(X, x), H)


is the Hurewicz map for any pointed space (X, x); and the cohomol-

ogy is singular cohomology (in particular it only depends on the weak

homotopy type of the space).

Now look at the pullback of this class

ℜ(h)∗(η) ∈ H3(ℜ(A), H).

The hypothesis that ℜ(u) induces an isomorphism on π3 implies that

Hur(ℜ(h)∗(η)) : π3(ℜ(A), r(a))∼=→ H.

In particular,Hur(ℜ(h)∗(η)) is nonzero so ℜ(h)∗(η) is nonzero inH3(ℜ(A), H).

This is a contradiction because ℜ(A) is weak homotopy-equivalent to the

3-type of S2, and H = Z, but H3(S2,Z) = 0.

This contradiction completes the proof of the theorem.

As was discussed in [108], this result motivates the search for a notion

of higher category weaker than the notion of strict n-category. Following

the yoga described by Lewis [151], it appears to be sufficient to weaken

any single particular aspect.

5

Simplicial approaches

There are a number of approaches to weak higher categories based on

the simplicial category ∆, including the Segal approach and its itera-

tions which are the main subject of our book. We also discuss several

other approaches, which concern first and foremost the theory of (∞, 1)-

categories.

5.1 Strict simplicial categories

A simplicial category is a K -enriched category. It has a set of objects

Ob(A), and for each pair x, y ∈ Ob(A) a simplicial set A(x, y) thought

of as the “space of morphisms” from x to y. The composition maps are

morphisms of simplicial sets A(x, y) ×A(y, z) → A(x, z) satisfying the

associativity condition strictly, that is for any x, y, z, w the diagram of

simplicial sets

A(x, y)×A(y, z)×A(z, w) → A(x, y) ×A(y, w)

A(x, z)×A(z, w)

↓

→ A(x,w)

↓

commutes. The identities of A are points (i.e. vertices) of the simpli-

cial sets A(x, x) satisfying left and right identity conditions which are

equalities of maps A(x, y) → A(x, y).

A functor of simplicial categories f : A → B consists of a map f :

Ob(A) → Ob(B) and for each x, y ∈ Ob(A), a map of simplicial sets

fx,y : A(x, y) → B(f(x), f(y)), compatible with the composition maps

and identities in an obvious way. In keeping with our general notation


70 Simplicial approaches

for enriched categories, the category of simplicial categories is denoted

Cat(K ).

Given a simplicial category A, we define its truncation τ≤1(A) to be

the category whose set of objects is the same as Ob(A), but for any

x, y ∈ Ob(A)

τ≤1(A)(x, y) := π0(A(x, y)).

The composition maps and identities for A define composition maps and

identities for τ≤1(A), and we obtain a functor

τ≤1 : Cat(K ) → Cat.

A functor f : A → B between simplicial categories is said to be fully

faithful if for every x, y ∈ Ob(A) the map fx,y : A(x, y) → B(f(x), f(y))

is a weak equivalence of simplicial sets, in other words a weak equiva-

lence in the standard model structure of K . A functor f is said to be

essentially surjective if the functor τ≤1(f) between usual 1-categories is

essentially surjective, in other words it induces a surjection on sets of

isomorphism classes Isoτ≤1(A) ։ Isoτ≤1(B). A functor f : A → B is

said to be a Dwyer-Kan equivalence between simplicial categories, if it

is fully faithful and essentially surjective. In this case, τ≤1(f) is also an

equivalence of categories, in particular it is bijective on sets of isomor-

phism classes.

Given a simplicial category A, its underlying category is the category

with objects Ob(A), but the morphisms from x to y are the set of points

or vertices of the simplicial set A(x, y). This is not to be confused with

τ≤1(A), but there is a natural projection functor from the underlying

category to the truncated category. An “arrow” in A from x to y means

a map in this underlying category; such an arrow is said to be an internal

equivalence if it projects to an isomorphism in τ≤1(A). In these terms,

the essential surjectivity condition for a functor f : A → B may be

rephrased as saying that any object of B is internally equivalent to the

image of an object of A.

Dwyer, Kan and Bergner have constructed a model category structure

on Cat(K ) such that the weak equivalences are the Dwyer-Kan equiva-

lences, and the fibrations are the functors f : A → B such that each fx,yis a fibration of simplicial sets, and furthermore f satisfies an additional

lifting condition which basically says that an internal equivalence in B

should lift to A if one of its endpoints lifts.

It is interesting to note that Dwyer and Kan started first by con-

structing a model structure on Cat(X,K ), the category of simplicial


categories with a fixed set of objects X . Refer to their papers [89] [90]

[91]. We will also adopt this route, following a suggestion by Clark Bar-

wick.

Simplicial categories appear in an important way in homotopy the-

ory. Quillen defined the notion of simplicial model category, and of N is

a simplicial model category then we obtain a simplicial category Nsplcf

of fibrant and cofibrant objects, such that its truncation τ≤1(Nsplcf ) ∼=

ho(N ) is the homotopy category of N . Dwyer and Kan then developped

the theory of simplicial localization which gives a good simplicial cate-

gory even when N doesn’t have a simplicial model structure. If C is any

category, and if we are given a subcollection of arrows Σ ⊂ Arr(C ), then

Dwyer and Kan define a simplicial category L(C ,Σ) whose truncation

is the classical Gabriel-Zisman localization: τ≤1(L(C ,Σ)) ∼= Σ−1(C ). In

the case where N is a simplicial model category, then the two options

L(N ,Nw) (where Nw denotes the class of weak equivalences) and Nsplcf ,

are Dwyer-Kan equivalent as simplicial categories [90] [91].

Even though simplicial categories have strictly associative composi-

tion, they are weaker than strict n-categories in the sense that the higher

categorical structure is encoded by the simplicial morphism sets rather

than by strict n−1-categories. Hence, the need for weak composition de-

scribed in the previous chapter, is not contradicatory with the fact that

strict simplicial categories model all (∞, 1)-categories. For the weaker

versions to be discussed next, one can rectify back to a strict simpli-

cial category, as was originally shown by Dwyer-Kan-Smith [92] and

Schwanzl-Vogt [184], then extended to Quillen equivalences between the

corresponding model structures by Bergner [34].

5.2 Segal’s delooping machine

The best-known version of Segal’s theory is his notion of infinite deloop-

ing machine or Γ-space. Grothendieck mentioned some correspondence

from Larry Breen in 1975 concerning this idea:

Dear Larry,. . . The construction which you propose for the notion of a non-strict n-

category, and of the nerve functor, has certainly the merit of existing, and ofbeing a first precise approach . . .

Otherwise, not having understood the idea of Segal in your last letter. . .

In the first letter of 1983, Grothendieck also mentioned the notion of

multisimplicial nerve of a (strict) n-category. So it would seem that the


idea of applying Segal’s delooping machine much as was done in [206],

was present in some sense at the time.

The starting point is Segal’s 1-delooping machine. Recall from topol-

ogy that for a pointed space (X, x) the loop space ΩX is the space of

pointed loops (S1, 0) → (X, x). These can be composed by reparametriz-

ing the loops in a well-known way, although the resulting composition

is only associative and unital up to homotopy. It is possible to replace

ΩX by a topological group, for example with Quillen’s realization of

homotopy types as coming from simplicial groups. A classifying space

construction usually denoted B(·) allows one to get back to the original

space:

B(ΩX) ∼ X.

A popular question in topology in the 1960’s was how to define various

types of structure on spaces homotopy equivalent to ΩX , which would

be weaker than the strong structure of topological group, but which

would include sufficiently much homotopical data to let us get back to

X by a classifying space construction B(·). Such a kind of structure

was known as a “delooping machine”. There were a number of examples

includingA∞-algebras (in the linearized case), PROP’s, operads, and the

one which we will be considering: Segal’s simplicial delooping machine.

Any of these delooping machines should lead to one or several notions

of higher category, indeed this has been the case as we shall discuss

elsewhere.

Let ∆ denote the simplicial category whose objects are denoted m

for positive integers m, and where the morphisms p → m are the (not-

necessarily strictly) order-preserving maps

0, 1, . . . , p → 0, 1, . . . ,m.

A morphism 1 → m sending 0 to i − 1 and 1 to i is called a principal

edge of m. A morphism which is not injective is called a degeneracy.

A simplicial set A : ∆o → Set such that A0 = ∗ and such that the

Segal maps obtained by the principal edges (01), (12), . . . , ((n − 1)n) ⊂

(0123 · · ·n) = [n]

Am → A1 × · · · × A1


are isomorphisms of sets, corresponds to a structure of monoid on the

set A1. Indeed, the diagram

A2

∼=→ A1 ×A1

A1

↓

where the horizontal map is the Segal map and the vertical map is

given by the third edge (02) ⊂ (012), provides a composition law A1 ×

A1 → A1. The degeneracy map A0 → A1 provides a unit—proved using

the degeneracy maps for A2—and consideration of A3 gives the proof of

associativity.

In Segal’s 1-delooping theory, this characterization of monoids is weak-

ened by replacing the condition of isomorphism by the condition of weak

homotopy equivalences (i.e. maps inducing isomorphisms on the πi).

Thus, a loop space is defined to be a simplicial space

A· : ∆o → Top

such that A0 is a single point, such that the Segal maps, again using the

principal edges (01), (12), . . . , ((n− 1)n) ⊂ (0123 · · ·n) = [n]

Am → A1 × · · · × A1

are weak homotopy equivalences, and which is grouplike in that the

monoid which results when we compose

π0 A· : ∆o → Top → Set

should be a group. Segal explains in [187] how to deloop such an object:

if Top is replaced by the category of simplicial sets then the structure

A· is a bisimplicial set, and its delooping B(A·) is just the diagonal

realization.

As was well known at the time, the characterization of monoids gen-

eralizes to give a characterization of the nerve of a category in terms

of Segal maps being isomorphisms. Indeed, a monoid can be viewed as

a category with a single object, and a small change in the definition

makes it apply to the case of categories with an arbitrary set of objects:

a simplicial set

A· : ∆o → Set


is the nerve of a 1-category if and only if the Segal maps made using

fiber products are isomorphisms

Am∼=→ A1 ×A0 · · · ×A0 A1.

Here the fiber products are taken over the two maps A1 → A0 corre-

sponding to (0) ⊂ (01) and (1) ⊂ (01), alternatingly and starting with

(1) ⊂ (01). These correspond to the inclusions of the intersections of

adjacent principal edges.

5.3 Segal categories

A Segal precategory is a bisimplicial set

A = Ap,k, p, k ∈ ∆

in other words a functor A : ∆o × ∆o → Set satisfying the globular

condition that the simplicial set k 7→ A0,k is constant equal to a set

which we denote by A0 (called the set of objects).

If A is a Segal precategory then for p ≥ 1 we obtain a simplicial set

k 7→ Ap,k

which we denote by Ap/. This yields a simplicial collection of simplicial

sets, or a functor ∆o → K to the Kan-Quillen model category K

of simplicial sets. One could instead look at simplicial spaces, that is

functors ∆o → Top such that A0 is a discrete space thought of as a set.

This gives an equivalent theory, although there are degeneracy problems

which apparently need to be treated in an appendix in that case [167]

[208]. We often use the “simplicial space” point of view when speaking

informally, as it is more intuitively compelling; however, we don’t want

to get into details of defining a model structure on Top, and instead use

K for technical statements.

For each m ≥ 2 there is a morphism of simplicial sets whose com-

ponents are given by the principal edges of m, which we call the Segal

map:

Am/ → A1/ ×A0 . . .×A0 A1/.

The morphisms in the fiber product A1/ → A0 are alternatively the

inclusions 0 → 1 sending 0 to the object 1, or to the object 0.

We would like to think of the inverse image A1/(x, y) of a pair (x, y) ∈


A0×A0 by the two maps A1/ → A0 referred to above, as the simplicial

set of maps from x to y.

We say that a Segal precategory A is a Segal category if for all m ≥ 2

the Segal maps

Am/ → A1/ ×A0 . . .×A0 A1/.

are weak equivalences of simplicial sets. This notion was introduced by

Dwyer, Kan and Smith [92] and Schwanzl and Vogt [184].

Given a strict simplicial category A, we obtain a corresponding Segal

precategory by setting

An/ :=∐

(x0,...,xn)∈Ob(A)n+1

A(x0, x1)× · · · × A(xn−1, xn).

This is a Segal category, because the Segal maps are isomorphisms. In

the other direction, a Segal category such that the Segal maps are iso-

morphisms comes from a unique strict simplicial category.

The “generators and relations” operation introduced in Chapter 16 is

a way of starting with a Segal precategory and enforcing the condition of

becoming a Segal category, by forcing the condition of weak equivalence

on the Segal maps. As a general matter we will call operations of this

type A 7→ Seg(A).

Suppose A is a Segal category. Then the simplicial set p 7→ π0(Ap/)

is the nerve of a category which we call τ≤1A. We say that A is a Segal

groupoid if τ≤1A is a groupoid. This means that the 1-morphisms of A

are invertible up to equivalence.

In fact we can make the same definition even for a Segal precategory

A: we define τ≤1A to be the simplicial set p 7→ π0(Ap/).

We can now describe exactly the situation envisaged in [3] [187]: a

Segal category A with only one object, A0 = ∗. We call this a Segal

monoid. If A is a groupoid then the homotopy theorists’ terminology is

to say that it is grouplike.

5.3.1 Equivalences of Segal categories

The basic intuition is to think of Segal categories as the natural weak

version of the notion of topological (i.e. Top-enriched) category. One of

the main concepts in category theory is that of a functor which is an

“equivalence of categories”. This may be generalized to Segal categories.

For simplicial (i.e. K -enriched) categories, this notion is due to Dwyer

and Kan, and is often called DK-equivalence. The same thing in the


context of n-categories is well-known (see Kapranov-Voevodsky [136] for

example); in the weak case it is described in Tamsamani’s paper [206].

We say that a morphism f : A → B of Segal categories is an equiva-

lence if it is fully faithful, meaning that for x, y ∈ A0 the map

A1/(x, y) → B1/(f(x), f(y))

is a weak equivalence of simplicial sets; and essentially surjective, mean-

ing that the induced functor of categories

τ≤1(A)τ≤1f→ τ≤1(B)

is surjective on isomorphism classes of objects. Note that this induced

functor τ≤1f will be an equivalence of categories as a consequence of the

fully faithful condition.

The homotopy theory that we are interested in is that of the category

of Segal categories modulo the above notion of equivalence. In particular,

when we search for the “right answer” to a question, it is only up to the

above type of equivalence. Of course when dealing with Segal categories

having only one object (as will actually be the case in what follows)

then the essentially surjective condition is vacuous and the fully faithful

condition just amounts to equivalence on the level of the“underlying

space” A1/.

In order to have an appropriately reasonable point of view on the ho-

motopy theory of Segal categories one should look at the closed model

structure (which is one of our main goals, specialized to the model cat-

egory K , see Chapter 17): the right notion of weak morphism from A

to B is that of a morphism from A to B′ where B → B′ is a fibrant

replacement of B.

5.3.2 Segal’s theorem

We define the realization of a Segal category A to be the space |A| which

is the realization of the bisimplicial set A. Suppose A0 = ∗. Then we

have a morphism

|A1/| × [0, 1] → |A|

giving a morphism

|A1/| → Ω|A|.

The notation |A1/| means the realization of the simplicial set A1/ and

Ω|A| is the loop space based at the basepoint ∗ = A0.


Theorem 5.3.1 (G. Segal [187], Proposition 1.5) Suppose A is a Segal

groupoid with one object. Then the morphism

|A1/| → Ω|A|.

is a weak equivalence of spaces.

Refer to Segal’s paper, or also May ([166] 8.7), for a proof. Tamsamani

noted that the same works in the case of many objects, and indeed this

was a key step in his proof of the topological realization theorem for

n-categories.

Corollary 5.3.2 Suppose A is a Segal groupoid. Then the morphism

|A1/| → Ω|A|.

is a weak equivalence in K .

Proof [206].

In order to do these things inside the world of simplicial spaces, the

additional cofibrancy conditions in Top would necessitate a discussion of

“whiskering” as is standard in delooping and classifying space construc-

tions (cf [187] [165], [167], [208]). This is why we have replaced “spaces”

by “simplicial sets” in the above discussion, and corresponds also to our

use of Reedy model structures in the main chapters.

5.3.3 (∞, 1)-categories

Simplicial categories and Segal categories are two models for what Lurie

calls the notion of (∞, 1)-category, meaning ∞-categories where the i-

morphisms are invertible (analogous to being an inner equivalence) for

i ≥ 2. The Dwyer-Kan simplicial localization may be viewed as the

localization in the (∞, 2)-category of (∞, 1)-categories. Part of our goal

in this book is to develop algebraic formalism useful for looking at these

situations, as well as their iterative counterparts for (∞, n)-categories.

5.3.4 Iteration

A subtle point is that simplicial categories don’t behave well under di-

rect products: the Dwyer-Kan-Bergner model structure on Cat(K ) is

not cartesian for the direct product because a product of two cofibrant

objects is no longer cofibrant. Thus, if we try to continue by looking at

Cat(Cat(K )) the resulting theory doesn’t have the right properties.


This hooks up with what we have seen in Chapter 4, that iterating the

strict enriched category construction, doesn’t lead to enough objects.

If we use Segal’s method, on the other hand, one can iterate the con-

struction with a better effect. This leads to Tamsamani’s iterative defi-

nition of n-categories. See Chapter 7. It is also related to Dunn’s theory

of iterated n-fold Segal delooping machines [85], and it will undoubtedly

be profitable to compare [206] and [85].

The next two sections will be devoted to brief descriptions of two

other major points of view on (∞, 1)-categories. After that, we discuss

the comparison between the various theories.

5.3.5 Strictification and Bergner’s comparison result

The various models of (∞, 1)-categories discussed above all furnish essen-

tially the same homotopy theory. Such a rectification result was known

very early for homotopy monoid structures. For Segal categories, the

first result of this kind was due to Dwyer, Kan and Smith who showed

how to rectify a Segal category into a strict simplicial category in [92].

Similarly, Scwanzl and Vogt showed the same thing in their paper in-

troducing Segal categories [184]. The full homotopy equivalence result,

stating that the rectification operation is a Quillen equivalence between

model structures, was shown by Bergner [34] at the same time as she

constructed the model structures in question. The model structure for

Segal categories is the special case M = K of the global construction

we are doing in the present book.

With respect to the models we are going to discuss next, Bergner also

gave a Quillen equivalence with Rezk’s model category of complete Segal

spaces, and the comparison can be extended to quasicategories too, as

shown by Joyal and Tierney in [130].

5.4 Rezk categories

Rezk has given a different way of using the Segal maps to specify an

(∞, 1)-categorical structure. Barwick showed how to iterate this con-

struction, and this iteration has now also been taken up by Lurie and

Rezk. Their iteration is philosophically similar to what we are doing in

the main part of this book. In the present section we discuss Rezk’s orig-

inal case, which he called “complete Segal spaces”. These objects enter

into Bergner’s three-way comparison [34].

5.4 Rezk categories 79

It will be convenient to start our discussion by refering to a generic

notion of (∞, 1)-category, which could be concretized by simplicial cat-

egories, or Segal categories. Recall that an (∞, 1)-category A has an

(∞, 0)-category or ∞-groupoid, as its interior denoted Aint. The inte-

rior is the universal ∞-groupoid mapping to A. For any x, y ∈ Ob(A),

the mapping space is the subspace Aint(x, y) ⊂ A(x, y) union of all the

connected components corresponding to maps which are invertible up

to equivalence. By Segal’s theorem (which will be discussed further in

Chapter 17), this corresponds to a space which we can denote by |Aint|.

It is the “moduli space of objects of A up to equivalence”: there is a

separate connected component for each equivalence class of objects. The

vertices coming from the 0-simplices correspond to the original objects

Ob(A), and within a connected component the space of paths from one

vertex to another, is the space Aint(x, y) of equivalences between the

corresponding objects.

In Rezk’s theory, our (∞, 1)-category is represented by a simplicial

space AR with AR0 = |Aint| in degree zero. The homotopy fiber of the

map

AR1 → AR0 ×A

R0

over a point (x, y) is (canonically equivalent to) the space of morphisms

A(x, y). The categorical structure is defined by imposing a Segal con-

dition on homotopy fiber products: for any n, there is a version of the

Segal map going to the homotopy fiber product

ARn → AR1 ×

hAR

0AR1 ×

hAR

0· · · ×hAR

0AR1

and this is required to be a weak equivalence. A complete Segal space

is a simplicial space satisfying these Segal conditions, and also the com-

pleteness condition which corresponds to the requirement AR0 = |Aint|.

We had formulated that requirement by first considering a generic the-

ory of (∞, 1)-categories. Internally to Rezk’s theory, the completeness

condition says that AR,int1 → AR0 ×AR0 should be equivalent to the path

space fibration, where AR,int1 ⊂ AR1 denotes the union of connected com-

ponents corresponding to morphisms which are invertible up to equiv-

alence. This condition is shown to be equivalent to a more abstract

condition useful for manipulating the model structure, see [178, 6.4] [34,

3.7] [130, Section 4].

Rezk’s theory is a little bit more complicated in its initial stages than

the theory of Segal categories. The Segal maps go to a homotopy fiber

product, which nevertheless can be assumed to be a regular fiber prod-


uct by imposing a Reedy fibrant condition on the simplicial space, for

example. Since the set of objects is not really too well-defined, the kind

of reasoning which we are considering here (and which was also followed

by Dwyer and Kan in their series of papers), breaking up the problem

into first a problem for higher categories with a fixed set of objects, then

varying the set of objects, is less available.

On the other hand, Rezk’s theory has the advantage that AR is a

canonical model for A up to levelwise homotopy equivalence in the cat-

egory of diagrams ∆o → Top. Thus, a map AR → BR of complete

Segal spaces, is an equivalence if and only if each ARn → BRn is a weak

equivalence of spaces (and it suffices to check n = 0 and n = 1 because

of the Segal conditions). This contrasts with the case of Segal categories,

where the set of objects A0 = Ob(A) is not invariant under equivalences

of categories.

As Bergner has pointed out [40], the canonical nature of the spaces in-

volved makes Rezk’s theory particularly amenable to calculating limits.

For example, if

AR → BR ← CR

are two arrows between complete Segal spaces, then the levelwise homo-

topy fiber product

URn := ARn ×hBR

nCRn

is again a complete Segal space, and it is the right homotopy fiber prod-

uct in the world of (∞, 1)-categories.

This again contrasts with the case of Segal categories, or indeed even

usual 1-categories. For example, letting E denote the 1-category with

two isomorphic objects υ0 and υ1, the inclusion maps

υ0 → E ← υ1

are equivalences of categories, so the homotopy fiber product in any rea-

sonable model structure for 1-categories, should also be equivalent to a

discrete singleton category. However, the fiber product of categories, or

of simplicial sets (the nerves) is empty. In Rezk’s theory, the degree 0

space ER0 will again be contractible, since there is only a single equiva-

lence class of objects of E and the have no nontrivial automorphisms.

As usual, for treating the technical aspects of the theory it is better to

look at bisimplicial sets rather than simplicial spaces. Rezk constructs

a model structure on the category of bisimplicial sets, such that the

fibrant objects are complete Segal spaces which are Reedy fibrant as

5.5 Quasicategories 81

∆o-diagrams and levelwise fibrant [178]. Bergner considers further this

theory and shows the equivalence with simplicial categories and Segal

categories [34].

Barwick has suggested to iterate this construction to a Rezk-style

theory of (∞, n) categories for all n, and Rezk has taken this up in [179].

He shows that the resulting model categories are cartesian, in particular

this gives a construction of the (∞, n+1)-category of (∞, n)-categories.

5.5 Quasicategories

Joyal and Lurie have developped extensively the theory of quasicate-

gories. These first appeared in the book of Boardman and Vogt [42]

under the name “restricted Kan complexes”. An important example ap-

peared in work of Cordier and Porter [74].

A good place to start is to recall Kan’s original horn-filling conditions

for the category of simplicial sets K . As K is a category of diagrams

∆o → Set, we have in particular the representable diagrams which we

shall denote R(n), defined by R(n)m := ∆([m], [n]). This is the “stan-

dard n-simplex”, classically denoted by R(n) = ∆[n]. For our purposes

this classical notation would seem to risk some confusion with too many

symbols ∆ around, so we call it R(n) instead. Now, R(n) has a stan-

dard simplicial subset denoted ∂R(n), which is the “boundary”. It can

be defined as the n − 1-skeleton of R(n), or as the union of the n − 1-

dimensional faces of R(n). The faces are indexed by 0 ≤ k ≤ n; in

terms of linearly ordered sets, the k-th face corresponds to the linearly

ordered subset of [n] obtained by crossing out the k-th element. Now,

the k-th horn Λ(n, k) is the subset of ∂R(n) which is the union of all the

n− 1-dimensional faces except the k-th one.

If X ∈ K is a simplicial set, then the universal property of the rep-

resentable R(n) says that Xn = HomK (R(n),X ). Kan’s horn-filling

condition says that any map Λ(n, k) → X extends to a map R(n) → X .

The simplicial sets X satisfying this condition are the fibrant objects of

the model structure on K .

Boardman and Vogt introduced the restricted Kan condition, satis-

fied by a simplicial set X whenever any map Λ(n, k) → X extends to

R(n) → X , for each 0 < k < n. In other words, they consider only the

horns obtained by taking out any except for the first and last faces.

This condition corresponds to keeping a directionality of the 1-cells in

X . This may be seen most clearly by looking at the case n = 2. A 2-cell


may be drawn as

R(2) :r

r

r

h g

f

where h, g and f are the 1-cells corresponding to edges (01), (12) and

(02) respectively. Such a 2-cell is thought of as the relation f = gh. In

the usual Kan condition, there are three horns which need to be filled:

Λ(2, 0) :r

r

r

h ?

f

Λ(2, 1) :r

r

r

h g

?

Λ(2, 2) :r

r

r

? g

f

However, in the restricted Kan condition, only the middle horn Λ(2, 1) is

required to be filled. This corresponds to saying that for any composable

arrows g and h, there is a composition f = gh. On the other hand, filling

the horn Λ(2, 0) would correspond to saying that given f and h, there is

g such that f = gh, which essentially means we look for g = fh−1; and

filling Λ(2, 2) would correspond to saying that given f and g there is h

such that f = gh, that is h = g−1f .

When we look at things in this way, it is clear that the full Kan condi-

tion corresponds to imposing, in addition to the categorical composition

of arrows, some kind of groupoid condition of existence of inverses. It

isn’t surprising, then, that Kan complexes correspond to ∞-groupoids.

Following through this philosophy has led Joyal to the theory of qua-

sicategories, which are simplicial sets satisfying the restricted Kan con-

5.6 Going between Segal categories and n-categories 83

dition, but viewed as (∞, 1)-categories with arrows which are not nec-

essarily invertible.

Making the translation from restricted Kan simplicial sets to (∞, 1)-

categories is not altogether trivial, most notably for any two vertices

x, y of a quasicategory X we need to define the simplicial mapping space

X (x, y); one possibility is to say that it is the Kan simplicial set

k 7→ Homx,y(R(1)×R(k),X )

where the superscript indicates maps sending 0×R(k) to x and 1×R(k)

to y. There is also a way of describing directly a simplicial category which

is the rectification of the corresponding (∞, 1)-category; see [180] for a

detailed discussion.

Joyal constructs a model category structure whose underlying cate-

gory is that of simplicial sets, in for which the fibrant objects are exactly

those satisfying the restricted Kan condition. The passage from a general

simplicial set to its fibrant replacement, done by enforcing the restricted

Kan horn filling conditions using the small object argument, is a version

of the “calculus of generators and relations” very similar to what we will

be discussing in Chapters 16 and 17 for the case of Segal categories.

In the three basic kinds of simplicial objects which we now have repre-

senting (∞, 1)-categories with weak composition, we can see a trade-off

between information content and simplicity. The simplest model is that

of quasicategories, which are just simplicial sets satisfying a very classi-

cal horn-filling condition; but in this case it isn’t easy to get back some

of the main pieces of information in an (∞, 1)-category such as the sim-

plicial mapping sets. At the other end, in Rezk’s complete Segal spaces,

the full information of the ∞-groupoid interior is contained within the

object, to the extent that the homotopy type of the ∆o-diagram is an

invariant of the (∞, 1)-category up to equivalence; on the other hand,

the initial steps of the theory are more complicated. The theory of Segal

categories fits in between: a Segal category has more information readily

at hand than a quasicategory, but less than a complete Segal space; and

the initial theory is more complicated than for quasicategories but less

than for complete Segal spaces.

5.6 Going between Segal categories and n-categories

We mention briefly the relationship between the notions of Segal cate-

gory and n-category. Tamsamani’s definition of n-category is recursive.


The basic idea is to use the same definition as above for Segal category,

but where the Ap/ are themselves n − 1-categories. The appropriate

condition on the Segal maps is the condition of equivalence of n − 1-

categories, which in turn is defined (inductively) in the same way as the

notion of equivalence of Segal categories explained above.

Tamsamani shows that the homotopy category of n-groupoids is the

same as that of n-truncated spaces. The two relevant functors are the

realization and Poincare n-groupoid Πn functors. Applying this to the

n−1-categories Ap/ we obtain the following relationship. An n-category

A is said to be 1-groupic (notation introduced in [194]) if the Ap/ are

n − 1-groupoids. In this case, replacing the Ap/ by their realizations

|Ap/| we obtain a simplicial space which satisfies the Segal condition.

Conversely if Ap/ are spaces or simplicial sets then replacing them by

their Πn−1(Ap/) we obtain a simplicial collection of n − 1-categories,

again satisfying the Segal condition. These constructions are not quite

inverses because

|Πn−1(Ap/)| = τ≤n−1(Ap/)

is the Postnikov truncation. If we think (heuristically) of setting n =∞

then we get inverse constructions. Thus—in a sense which I will not

currently make more precise than the above discussion—one can say

that Segal categories are the same thing as 1-groupic ∞-categories.

The passage from simplicial sets to Segal categories is the same as

the inductive passage from n− 1-categories to n-categories. In [193] was

introduced the notion of n-precat, the analogue of the above Segal precat.

Noticing that the results and arguments in [193] are basically organized

into one gigantic inductive step passing from n− 1-precats to n-precats,

the same step applied only once works to give the analogous results in

the passage from simplicial sets to Segal precats.

The notion of Segal category thus presents, from a technical point of

view, an aspect of a “baby” version of the notion of n-category. On the

other hand, it allows a first introduction of homotopy going all the way

up to∞ (i.e. it allows us to avoid the n-truncation inherent in the notion

of n-category).

One can easily imagine combining the two into a notion of “Segal n-

category” which would be an n-simplicial simplicial set satisfying the

globular condition at each stage. It is interesting and historically im-

portant to note that the notion of Segal n-category with only one i-

morphism for each i ≤ n, is the same thing as the notion of n-fold deloop-

ing machine. This translation comes out of Dunn [85], which apparently

5.7 Towards weak ∞-categories 85

dates essentially back to 1984. In retrospect it is not too hard to see

how to go from Dunn’s notion of En-machine, to Tamsamani’s notion of

n-category, simply by relaxing the conditions of having only one object.

Metaphorically, n-fold delooping machines correspond to the Whitehead

tower, whereas n-groupoids correspond to the Postnikov tower.

There are other proposals for simplicial models for n-categories which

we haven’t been able to discuss. For example, Street proposed a model

based on simplicial sets with certain distinguished simplicial subsets

which he calls “thin subcomplexes”.

5.7 Towards weak ∞-categories

We mention here some ideas for going towards a theory of∞-categories.

As the iterative approach makes clear, there is no direct generalization

of our theory to the case n =∞ (which means the first infinite ordinal).

The notion of equivalence of Segal categories, crucial to everything, is

defined by a top-down induction, so by its nature it is related to some

kind of n-categories. As our procedure makes clear, and as came out in

[117] and [171], this iteration can start with any model category such

as K , allowing us to define Segal n-categories which in Lurie’s notation

correspond to (∞, n)-categories i.e. ∞-categories where all morphisms

are invertible starting from degree n+ 1.

Cheng’s argument [64] shows that if A is to be an∞-category with du-

als at all levels then, in an algebraic sense, all morphisms look invertible.

However, it is clear from Baez-Dolan’s predictions about the theory, that

we don’t want to identify ∞-categories with duals and ∞-groupoids, in-

deed they represent somehow complementary points of view, the first

being related to quantum field theory and the second to topology. Her-

mida, Makkai and Power have discussed these issues in [114].

From these observations and thinking about specific kinds of examples,

the following idea emerges: in a true ∞-category A, the information

about which i-morphisms should be considered as invertible, should be

thought of as an additional structure beyond the algebraic structure of

some kind of weak multiplication operations. Going back to the strict

case, one can well imagine a strictly associative ∞-category A in which

any i-morphism u has a morphism going in the other direction v and

i + 1-morphisms uv → 1 and vu → 1. Then, we could either declare

all morphisms to be invertible, in which case v would be the inverse of

u since the i + 1-morphisms going to the identities would be invertible;


or alternatively we could declare that no morphisms (other than the

identities) are invertible.

Both choices would be reasonable, and would lead to different ∞-

categories sharing the same underlying algebraic structure A.

So, if we imagine a theory of weak ∞-categories in which the infor-

mation of which morphisms are invertible is somehow present then it

becomes reasonable to define the truncation operations τ≤n as in [206]

but going from weak ∞-categories to weak n-categories. Thus an ∞-

category A would lead to a compatible system of n-categories τ≤n(A)

for all n.

This suggests a definition: it appears that one should get the right the-

ory by taking a homotopy inverse limit of the theories of n-categories.

Jacob Lurie had mentioned something like this in correspondence some

time ago. Given the compatibility of the Rezk-Barwick theory with ho-

motopy limits [40], that might be specially adapted to this task.

One might alternatively be able to view the theory of∞-categories as

some kind of first “fixed point” of the operation M → PC(M ) which

we will construct in the main chapters.

We will leave these considerations on a speculative level for now, hop-

ing only that the techniques to be developed in the main part of the

book will be useful in attacking the problem of ∞-categories later.

6

Operadic approaches

Apart from the simplicial approaches, the other main direction is com-

prised of a number of operadic approches, definitions of higher categories

based on Peter May’s notion of “operad”. This dichotomy is not surpris-

ing, given that operads and simplicial objects are the two main ways

of doing delooping machines in algebraic topology. The operadic ap-

proaches are not the main subject of this book, so our presentation will

be more succinct designed to inform the reader of what is out there.

Tom Leinster’s book [149] has a very complete discussion of the rela-

tionship between operads and higher categories. His paper [148] gives a

brief but detailed exposition of numerous different definitions of higher

categories, including several in the operadic direction, and that was the

first appearence in print of some definitions such as Trimble’s for exam-

ple.

6.1 May’s delooping machine

We start by recalling Peter May’s delooping machine. An operad is a col-

lection of sets O(n), thought of as the “set of n-ary operations”, together

with some maps which are thought of as the result of substitutions:

ψ : O(m)×O(k1)× · · · ×O(km) → O(k1 + · · ·+ km)

for any uplets of integers (m; k1, . . . , km). If we think of an element

u ∈ O(n) as representing a function u(x1, . . . , xn) then ψ(u; v1, . . . , vm)

is the function of k1 + . . .+ km variables

u(v1(x1,...,xk1),v2(xk1+1,...,xkk1+k2

),...,vm(xk1+···+km−1+1,...,xk1+···+km )).


88 Operadic approaches

The substitution operation ψ is required to satisfy the appropriate ax-

ioms [165].

A topological operad is the same, but where the O(n) are topological

spaces; it is said to be contractible if each O(n) is contractible. More

generally, we can consider the notion of operad in any category admitting

finite products.

There is a notion of action of an operad O on a set X , which means an

association to each u ∈ O(n) of an actual n-ary functionX×· · ·×X → X

such that the substitution functions ψ map to function substitution as

described above.

If X is a space and O is a topological operad, we can require that

the action consist of continuous functions O(n) × Xn → X , or more

generally if O is an operad in M then an action of O on X ∈ M is

a collection of morphisms O(n) × Xn → X satisfying the appropriate

compatibility conditions.

A delooping structure on a space X , is an action of a contractible

operad on it.

The typical example is that of the little intervals operad: here O(n) is

the space of inclusions of n consecutive intervals into the given interval

[0, 1], and substitution is given by pasting in. A variant is used in Section

6.4 below.

6.2 Baez-Dolan’s definition

In Baez and Dolan’s approach, the notion of operad is first and foremost

used to determine the shapes of higher-dimensional cells. They introduce

a category of opetopes and a notion of opetopic set which, like the case

of simplicial sets, just means a presheaf on the category of opetopes.

They then impose filler conditions. Their scheme of filler conditions is

inductive on the dimension of the opetopes, but is rather intricate. We

describe the category of opetopes by drawing some of the standard pic-

tures, and then give an informal discussion of the filler conditions. In

addition to the original papers [6] [9], Leinster’s [148] was one of our

main sources. Readers may also consult [63] [142] for other approaches

to defining and calculating with opetopes.

An n-dimensional opetope should be thought of as a roughly globular

n-dimensional object, with an output face which is an n−1-dimensional

opetope, and an input face which is a pasting diagram of n−1-dimensional

6.2 Baez-Dolan’s definition 89

opetopes. To paste opetopes together, match up the output faces with

the different pieces of the input faces.

The only 0-dimensional opetope is a point. The only 1-dimensional

opetope has as input and output a single point, so it is a single arrow

r r-

A pasting diagram of 1-dimensional opetopes can therefore be composed

of several arrows joined head to tail:

r

r r

r HHHH*

-

j

A 2-dimensional opetope can then have such a pasting diagram as input,

with a single 1-dimensional opetope or arrow as output:

r

r r

r

⇓

3z

^:

A pasting diagram of 2-dimensional opetopes would arise if we add

on some other opetopes, with the output edges attached to the input

edges. This can be done recursively. A picture where we add three more

opetopes on the three inputs, with 2, 1 and 3 input edges respectively;

then a fourth one on the second input edge of the first new one, would

look like this:

r

r

r

3r

3-s rR- rz

rU

r^:

⇓

⇓⇓

⇓ ⇓

Now, a 3-dimensional opetope could have the above pasting diagram

as input; in that case, the output opetope is supposed to have the shape


of the boundary of the pasting diagram:

r

r

r

3r

srR rz

rU

r

⇓

That is to say, it is a 2-dimensional opetope with 7 input edges. We

don’t draw the 3-dimensional opetope; it should look like a “cushion”.

Baez and Dolan define in this way the category of opetopes Otp. The

reader is refered to the main references [6] [9] [114] [63] [142] [148] for

the precise definitions. A Baez-Dolan n-category is an opetopic set, that

is a functor

A : Otp → Set,

which is required to satisfy some conditions. Refer again to [6] [9] [114]

[63] [142] [148] for the precise statements of these conditions. see also

[217] for one of the most recent treatments.

Included in the conditions are things having to do with truncation so

that the cells of dimension > n + 1 don’t matter, when speaking of an

n-category.

Beyond this truncation, one of the main features of the Baez-Dolan

viewpoint is that an n-dimensional opetope represents an n-morphism

which is not necessarily invertible, from the “composition” of the opetopes

in the input face, to that of the output face. In particular, the output

face of an opetope is not necessarily the same as the composition of the

input faces; this distinguishes their setup from the Segal-style picture we

are mostly considering in the present book, in which a simplex represents

a composition with the outer edge being equivalent to the composition

of the principal edges.

The idea that opetopes represent arbitrary morphisms from the com-

position of the inputs, to the output, makes it somewhat complicated to

collect together and write down all of the appropriate conditions which

an opetopic set should satisfy in order to be an n-category. The main

step is to designate certain cells as “universal”, meaning that the output

edge is equivalent to the composition of the input edges, which is basi-

cally a homotopic initiality property. The necessary uniqueness has to

be treated up to equivalence, whence the need for a definition of equiv-


alence in the induction. All of this leads to Baez and Dolan’s notions of

niches, balanced cells and so forth.

As a rough approximation, we can say that these conditions are a form

of horn-filler conditions, generalizing the restricted Kan condition used

in the definition of quasicategory [42] [126] but adapted to the opetopic

context.

To close out this section, here are a few thoughts on the possible rela-

tionship between this theory and the many other theories of n-categories.

For one thing, the fact that the opetopic cells represent explicitly the

n-morphisms which are not necessarily invertible, would seem to render

this theory particularly well adapted to looking at things like lax func-

tors (what Benabou calls “morphisms” as opposed to “homomorphisms”

[28]).

On the other hand, for a comparison with other theories, it would be

interesting to investigate functors F : Otp → P where P is a closed

model category of “n-categories” (for example, the P = PCn(Set)

which we are going to be constructing in the rest of the book). Given

such a functor, if B ∈P is a fibrant object, we would obtain an opetopic

set F∗(B), and conversely given an opetopic set A we could construct

its realization F!(A) in P. Under the right hypotheses, one hopes, these

should set up a Quillen adjunction between a model category of opetopic

sets, and the other model category P.

If both of the above remarks could be realized, it would lead to the

introduction of powerful new techniques for treating lax functors in any

of the other theories of n-categories.

6.3 Batanin’s definition

Batanin’s definition is certainly the closest to Grothendieck’s original vi-

sion. Recall the passage that we have quoted from “Pursuing Stacks” on

p. 17 in Chapter 1 above, saying that whenever two morphisms which are

naturally obtained as some kind of composition, have the same source

and target, there should be a homotopy between them at one level up.

Batanin’s definition puts this into place, by carefully studying the notion

of possible composition of arrows in a higher category. It was recently

pointed out by Maltsiniotis that Grothendieck had in fact given a def-

inition of higher groupoid [161] and that a small modification of that

approach yields a definition of higher category which is very close to

Batanin’s [162].


Tom Leinster and Eugenia Cheng have refined Batanin’s original work,

and our discussion will be informed by their expositions [148] [149] [65],

to which the reader should refer for more details. Leinster has also intro-

duced some related definitions of weak n-category based on the notion

of multicategory, again for this we refer to [149].

One of Batanin’s innovations was to introduce a notion of operad

adapted to higher categories, based on the notion of globular set, a

presheaf on the category G which has objects gi for each i, and maps

s!, t! : gi → gi+1. The objects gi are supposed to represent globular

pictures of i-morphisms, for example g2 may be pictured as

r rR

⇓

and the maps s!, t! are viewed as inclusions of smaller-dimensional glob-

ules on the boundary. Thus, the inclusion maps are subject to the rela-

tions

s!s! = t!s!, t!s! = t!t!.

Dually, a globular set is a collection of sets Ai := A(Gi) with source

and target maps

Ais→t→ Ai−1 · · ·A1

s→

t→ A0

subject to the relations s s = s t and s t = t t.

This definition differs slightly from the one which was suggested in

Chapter 2; there we looked at what should probably be called a unital

globular set having identity maps going back in the other direction i :

Ai → Ai+1. For the present purposes, we consider globular sets without

identity maps.

Batanin’s idea is to use the notion of globular set to generate the

appropriate kind of “collection” for a notion of higher operad. One can

think of an operad as specifying a collection of operations of each pos-

sible arity; and an arity is a possible configuration of the collection of

inputs. For Batanin’s globular operads, an input configuration is rep-

resented by a globular pasting diagram P , and the family of operations

of arity P should itself form a globular set. A globular pasting diagram


in dimension n is an n-cell in the free strict ∞-category generated by a

single non-identity cell in each dimension.

Batanin gives an explicit description of the n-cells in this free ∞-

category, using planar trees. The nodes of the trees come from the

source and target maps between stages in a globular set, which is not

to be confused with the occurence of trees in parametrizations of ele-

ments of a free nonassociative algebra (where the nodes correspond to

parenthetizations)—for the free ∞-category involved here, it is strictly

associative and going up a level in the tree corresponds rather to going

from i-morphisms to i+ 1-morphisms.

As Cheng describes in a detailed example [65], a globular pasting

diagram is really just a picture of how one might compose together

various i-morphisms. Given any globular set A, and a globular pasting

diagram P , we get a set denoted AP consisting of all the ways of filling

in P with labels from the globular set A, consistent with sources and

targets.

Now, a globular operad B consists of specifying, for each globular

pasting diagram P , a globular set B(P ) of operations of arity P . This

globular set consists in particular of sets B(P )n for each n calle the set

P -ary operations of level n, together with sources and targets

B(P )n+1

s→t→ B(P )n

satisfying the globularity relations. An action of B on a globular set A

consists of specifying for each globular pasting diagram P of degree n,

a map of sets

B(P )n ×AP → An

such that the source diagram

B(P )n ×AP → An

B(sP )n−1 ×AsP

s

↓

→ An−1

s

↓

commutes and similarly for the target diagram. Here sP and tP are the

source and target of the pasting diagram P , which are pasting diagrams

of degree n − 1. Of course we need to describe an additional operadic

structure on B and the action should be compatible with this too, but

it is easier to first consider what data an algebra should have.


To describe what an operadic structure should mean, notice that there

is an operation of substitution of globular pasting diagrams, indeed the

free ∞-category on one cell in each dimension can be denoted GP, and

if P ∈ GPm is a globular pasting diagram in degree m then we get a

map of globular sets

GPP → GP,

that is to say that given a labeling of the cells of P where the labels Ljare themselves globular pasting diagrams, we can substitute the labels

into the cells and obtain a big resulting globular pasting diagram.

Now the operadic structure, which is in addition to the structure of

B described above, should say that for any element of GPP consisting

of labels denoted Lj ∈ GPnj for the cells of P and yielding an output

pasting diagram S, if we are given elements of the B(Lj) plus an element

of B(P ) then there should be a big output element of B(S). This needs

to be compatible with source and target operations, as well as compatible

with iteration in the style of usual operads.

Batanin describes explicitly the combinatorics of this using the iden-

tification between pasting diagrams and trees, whereas Leinster takes a

more abstract approach using monads and multicategories. We refer the

reader to the references for further details.

The main point is that this discussion establishes a language in which

to say that the system of coherencies should satisfy a globular con-

tractibility condition. Observe that contractibility is a very easily de-

fined property of a globular set, and it doesn’t depend on any kind of

composition law: a globular set A is contractible if it is nonempty, and

if for any two f, g ∈ Am with s(f) = s(g) and t(f) = t(g), there exists

an h ∈ Am+1 with s(h) = f and t(h) = g.

Note that this definition is really only appropriate if we are working

with globular sets which are truncated at some level (i.e. trivial above

a certain degree n), or ones which are supposed to represent (∞, n)-

categories, that is ones in which all i-morphisms are declared invertible

for i > n. For the purposes of Batanin’s definition of n-categories, this

is the case.

Now, a Batanin n-category is a globular set A provided with an ac-

tion of a globular operad B, such that B is contractible. Batanin con-

structs a universal globular operad, but it can also be convenient to work

with other contractible globular operads, as in Cheng’s comparison with

Trimble’s work which we discuss next.

The elements of B(P )n are the “natural operations taking a collection

6.4 Trimble’s definition and Cheng’s comparison 95

of morphisms of various degrees, and combining together to get an n-

morphism”. Contractibility of B really puts into effect Grothendieck’s

dictum that, given two natural operations f and g with the same source

and target, there should be an operation h one level higher whose source

is f and whose target is g. So, Batanin’s definition is the closest to what

Grothendieck was asking for.

6.4 Trimble’s definition and Cheng’s comparison

Trimble’s definition of higher category has acquired a central role be-

cause of Cheng’s recent work comparing it to Batanin’s definition [65].

This has also been taken up by Batanin, Cisinski and Weber in [25].

Trimble’s framework has the advantage that it is iterative. In the future

it should be possible to establish a comparison with the iterative Segal

approach we are discussing in the rest of the book. Such a comparison

result would be very interesting, but we don’t discuss it here. Instead

we just give the basic outlines of Trimble’s approach and state Cheng’s

comparison theorem. We are following very closely her article [65].

Consider the following operad OT in Top:

OT (n) ⊂ C0([0, 1], [0, n])

is the subset of endpoint-preserving continuous maps. The operad struc-

ture ψT is given by

ψ(f ; g1, . . . , gm)(t) := gj(f(t)− (j − 1)), f(t) ∈ [j − 1, j] ⊂ [0,m].

The spaces OT (n) are contractible. This topological operad is particu-

larly adapted to loop spaces and path spaces. If Z ∈ Top, and x, y ∈ Z

let Pathx,y(Z) denote the space of paths γ : [0, 1] → Z with γ(0) = x

and γ(1) = y. For any sequence of points x0, . . . , xm ∈ X we have a

“substitution” map

OT (m)× Pathx0,x1(Z)× · · · × Pathxn−1,xn(Z),

and these are compatible with the operad structure. In particular when

the points are all the same, this gives an action of OT on the loop space

Ωx(Z).

A Trimble topological category consists of a set X “of objects”, to-

gether with a collection of spaces A(x, y) for any x, y ∈ X , and collection

of maps

φx· : OT (n)×A(x0, x1)× · · · × A(xn−1, xn) → A(x0, xn)


for any sequence (x0, . . . , xn) in X . These should satisfy a compatibil-

ity condition with the operad structure ψT for OT : given a sequence

x0, . . . , xm and sequences yi0, . . . , yiki

with yi0 = xi−1 and yiki = xi,

φ(ψT (f ; g1, . . . , gm);u11, . . . , u1k1 , . . . , u

m1 , . . . , u

mkm)

= φ(f ;φ(g1;u11, . . . , u

1k1), . . . , φ(g1m;um1 , . . . , u

mkm).

Similarly, for any category M admitting finite products, and any op-

erad (O,ψ) in M , we can define a notion of (M , O, ψ)-category; this

consists of a set X of objects, together with a collection of A(x, y) ∈M

for any x, y ∈ X , and a collection of maps φ as above satisfying the same

compatibility condition.

Suppose we are given a category M with finite products, and a functor

Π : Top →M , then we obtain an operad Π(OT ) in M . We get the

notion of (M ,Π(OT ),Π(ψT ))-category. If contractibility makes sense in

M and Π(OT (n)) is contractible, then this is a generalization due to

Cheng [65], of Trimble’s notion of higher category enriched in M .

Trimble’s original definition, which first appeared publicly in [148], in-

cluded an inductive construction of the Poincare n-groupoid functor Πn.

He defines inductively a sequence of categories which Cheng denotes by

Vn, starting with V0 = Set; together with product-compatible Poincare

n-groupoid functors Πn : Top → Vn. The inductive definition is that

Vn+1 is the category of (Vn,Πn(OT ),Πn(ψ

T ))-categories, and

Πn+1(Z) = (X,A, φ)

where X := Zdisc is the discrete set of points of Z, where A is defined by

A(x, y) := Πn(Pathx,y(Z)), and φ is defined using the action described

above of OT on the path spaces. See [65] for further details, as well as

for the generalization to the case where an arbitrary contractible operad

Pn replaces Πn(OT ).

Cheng goes on to compare this family of definitions, with Batanin’s

definition: she shows how to combine the P0, P1, . . . , Pn−1 together to

form a contractible globular operad Q(n) such that the category of glob-

ular algebras of Q(n) is Vn. This expresses Vn as a category of Batanin

n-categories for this particular choice of contractible globular operad.

The reader is refered to [65], as well as to [30], [25] and [69] for related

aspects of this kind of comparison.

6.5 Weak units 97

6.5 Weak units

In the course of investigating the nonrealization of homotopy 3-types by

strict 3-groupoids ([197], see Chapter 4), the main obstruction seemed

to be the strict unit condition in a strict n-category. This is one of the

main aspects which allows the Eckmann-Hilton argument to work. That

was explained to me by Georges Maltsiniotis and Alain Bruguieres, but

has of course been well-known for a long time. This led to the conjec-

ture that maybe it would be sufficient to keep the strict associativity of

composition, but to weaken the unit condition.

Recall from homotopy theory (cf [151]) the yoga that it suffices to

weaken any one of the principal structures involved. Most weak notions

of n-category involve a weakening of the associativity, or eventually of

the Godement interchange conditions.

O. Leroy [150] and apparently, independantly, Joyal and Tierney [129]

were the first to do this in the context of 3-types. See also Gordon, Power,

Street [104] and Berger [29] for weak 3-categories and 3-types. Baues [26]

showed that 3-types correspond to quadratic modules (a generalization

of the notion of crossed complex) [26]. Then come the models for weak

higher categories which we are considering in the rest of the book.

It seems likely that the arguments of [136] would show that one could

instead weaken the condition of being unital, that is having identities,

and keep associativity and Godement. We give a proposed definition of

what this would mean and then state two conjectures.

This can be motivated by looking at the Moore loop space ΩxM (X)

of a space X based at x ∈ X , cited in [136] as a motivation for their

construction. Recall that ΩxM (X) is the space of pairs (r, γ) where r is

a real number r ≥ 0 and γ = [0, r] → X is a path starting and ending

at x. This has the advantage of being a strictly associative monoid. On

the other side of the coin, the “length” function

ℓ : ΩxM (X) → [0,∞) ⊂ R

has a special behavoir over r = 0. Note that over the open half-line

(0,∞) the length function ℓ is a fibration (even a fiber-space) with fiber

homeomorphic to the usual loop space. However, the fiber over r = 0

consists of a single point, the constant path [0, 0] → X based at x.

This additional point (which is the unit element of the monoid ΩxM (X))

doesn’t affect the topology of ΩxM (at least if X is locally contractible

at x) because it is glued in as a limit of paths which are more and more

concentrated in a neighborhood of x. However, the map ℓ is no longer


a fibration over a neighborhood of r = 0. This is a bit of a problem

because ΩxM is not compatible with direct products of the space X ; in

order to obtain a compatibility one has to take the fiber product over R

via the length function:

Ω(x,y)M (X × Y ) = ΩxM (X)×R ΩyM (Y ),

and the fact that ℓ is not a fibration could end up causing a problem in

an attempt to iteratively apply a construction like the Moore loop-space.

Things seem to get better if we restrict to

ΩxM ′(X) := ℓ−1((0,∞)) ⊂ ΩxM (X),

but this associative monoid no longer has a strict unit. Even so, the

constant path of any positive length gives a weak unit.

A motivation coming from a different direction was an observation

made by Tamsamani early in the course of his thesis work. He was try-

ing to define a strict 3-category 2Cat whose objects would be the strict

2-categories and whose morphisms would be the weak 2-functors be-

tween 2-categories (plus notions of weak natural transformations and

2-natural transformations). At some point he came to the conclusion

that one could adequately define 2Cat as a strict 3-category except that

he couldn’t get strict identities. Because of this problem we abandonned

the idea and looked toward weakly associative n-categories. In retrospect

it would be interesting to pursue Tamsamani’s construction of a strict

2Cat but with only weak identities.

In [197] was introduced a preliminary definition of weakly unital strict

n-category (called “snucategory” there), including a notion of direct

product. The proposed definition went as follows. Suppose we know what

these are for n−1. Then a weakly unital strict n-category C consists of a

set Ob(C) of objects together with, for every pair of objects x, y ∈ Ob(C)

a weakly unital strict n−1-category C(x, y) and composition morphisms

C(x, y)× C(y, z) → C(x, z)

which are strictly associative, such that a weak unital condition holds.

We now explain this condition. An element ex ∈ C(x, x) is called a weak

identity if:

—composition with e induces equivalences of weakly unital strict n− 1-

categories

C(x, y) → C(x, y), C(y, x) → C(y, x);

—and if e · e is equivalent to e. It would be best to complete this last

6.5 Weak units 99

condition to the fuller collection of coherence conditions introduced by

Kock [140].

In order to complete the recursive definition we must define the notion

of when a morphism of weakly unital strict n-categories is an equivalence,

and we must define what it means for two objects to be equivalent. A

morphism is said to be an equivalence if the induced morphisms on the

C(x, y) are equivalences of weakly unital strict n− 1-categories and if it

is essentially surjective on objects: each object in the target is equivalent

to the image of an object. It thus remains just to be seen what internal

equivalence of objects means. For this we introduce the truncations τ≤iC

of a weakly unital strict n-category C. Again this is done in the same

way as usual: τ≤iC is the weakly unital strict i-category with the same

objects as C and whose morphism i− 1-categories are the truncations

Homτ≤iC(x, y) := τ≤i−1C(x, y).

This works for i ≥ 1 by recurrence, and for i = 0 we define the trunca-

tion to be the set of isomorphism classes in τ≤1C. Note that truncation is

compatible with direct product (direct products are defined in the obvi-

ous way) and takes equivalences to equivalences. These statements used

recursively allow us to show that the truncations themselves satisfy the

weak unary condition. Finally, we say that two objects are equivalent if

they map to the same thing in τ≤0C.

Proceeding in the same way as in Chapter 2, we can define the notion

of weakly unital strict n-groupoid.

Conjecture 6.5.1 There are functors Πn and ℜ between the cate-

gories of weakly unital strict n-groupoids and n-truncated spaces (going

in the usual directions) together with adjunction morphisms inducing an

equivalence between the localization of weakly unital strict n-groupoids

by equivalences, and n-truncated spaces by weak equivalences.

Joachim Kock has developed the right definition of weakly unitary

strict n-category which takes into account the full collection of higher

coherence relations [140] [141] rather than just asking that e ∼ e · e; we

refer the reader there for his definition which supersedes the preliminary

version described above.

Joyal and Kock have proven Conjecture 6.5.1 for the case n = 3 in

[128]. For general n, one could hope to apply the argument of [136].

These results concern the case of groupoids, however we might also

expect that weakly unital strict n-categories serve to model all weak

n-categories:


Conjecture 6.5.2 The localization of the category of weakly unital

strict n-categories by equivalences, is equivalent to the localizations of

the categories of weak n-categories of Tamsamani and/or Baez-Dolan

and/or Batanin by equivalences.

While we’re discussing the subject of unitality conditions, the follow-

ing remark is in order. The role of strict unitality conditions in the

interchange or Eckmann-Hilton relations, and the consequent nonreal-

ization of homotopy types with nontrivial Whitehead bracket, suggests

that we need to take some care about this point in the general argument

which will be developped in Parts III and IV. It turns out that, in or-

der to insure a good cartesian property, our Segal-style weakly enriched

categories should nonetheless be endowed with strict units in a certain

sense. These correspond to the degeneracies in the simplicial category

∆, and are important for the Eilenberg-Zilber argument which yields

the cartesian property. They don’t correspond to full strict units in the

maximal possible way, because the composition operation will not even

be well-defined; that is why we will be able to impose the unitality con-

dition in Part III, without running up against the problems identified in

Chapter 4.

6.6 Other notions

The theory of n-categories is an essentially globular theory: an i-morphism

has a single i−1-morphism as source, and a single one as target. This ba-

sic shape can be relaxed in many ways. For example, Leinster and others

have investigated notions of multicategory where the input is a collection

of objects rather than just one object. This is somewhat related to the

opetopic shapes introduced by Baez and Dolan.

Another way of relaxing the globular shape is to iterate the internal

category construction. Brown and Loday constructed the first algebraic

representation for homotopy n-types, with the notion of Catn-group.

Let Cat1(Gp) denote the category of internal categories in the category

Gp, then Catn+1(Gp) is the category of internal categories in Catn(Gp).

There is a natural realization functor from Catn(Gp) to homotopy n-

types, and Brown and Loday prove that all homotopy n-types are real-

ized.

Paoli has recently refined this model to go back in the globular di-

rection, by introducing a notion of special Catn-group [170]. If we think

6.6 Other notions 101

of an internal category as being a pair of objects connected by several

morphisms, then an internal n-fold category may be seen as a collection

of objects arranged at the vertices of an n-cube. The speciality condition

requires that certain faces of the cube be contractible.

The speciality condition is a sort of weak globularity condition. An

internal n-fold category is not in and of itself a globular object, because

the object of objects may be nontrivial. A strict globular condition would

have the object of objects be a discrete set; the speciality condition

requires only that it be a disjoint union of contractible Catn−1-groups.

Paoli shows that the special Catn-groups model homotopy n-types,

and she relates this model to Tamsamani’s model. This provides a semistric-

tification result saying that we can have a strictly associative composi-

tion at any one stage of Tamsamani’s model. An alternative proof of this

result for semistrictification at the last stage, may be obtained using the

fact that Segal categories are equivalent to strict simplicial categories.

Paoli’s model relates to Lewis’s principle cited above [151] in an in-

teresting way: in a Catn-group all the structures—associativity, units,

inverses, interchange—are strict; the special Catn-groups weaken instead

the globularity condition itself.

Penon’s definition [172] is completely algebraic in the sense that a

weak n-category is an algebra over a monad. For a rapid description

of the monad, one can refer to [148, pp 14–17], see also [23] [66] [67]

[99]. Penon introduces a category Q consisting of arrows of “ω-magmas”

M → S where S is a strict ω-category, together with a contractive

structure on π. The monad is adjoint to the functor “underlying glob-

ular set of A♯”. If A is a globular set then it goes to an element of

Q with S being the free strict ω-category generated by A. Note that

this free category is the set of globular pasting diagrams as in Section

6.3, and M may be viewed as some sort of family of elements over S

with a contractible structure. In this sense Penon’s definition uses ob-

jects of the same sort as Batanin’s definition, indeed Batanin has made

a more precise comparison in [23]. Penon’s definition uses globular sets

with identities (“reflexive globular sets”) whereas Batanin’s were with-

out them, so [23] proposes a modified version of Penon’s definition with

non-reflexive globular sets. Cheng and Makkai have pointed out that

it is better to use the non-reflexive version, since the reflexive version

doesn’t lead to all the objects one would want [67], essentially because

of the Eckmann-Hilton argument. Futia proposes a generalized family of

Penon-style definitions in [99]. In these definitions, one could say that


globally the goal is to be able to parametrize higher compositions sorted

according to their shapes which are globular pasting diagrams.

7

Weak enrichment over a cartesian modelcategory: an introduction

To close out the first part of the book, we describe in this chapter the

basic outlines of the theory which will occupy the rest of the work. The

basic idea, already considered in Pelissier’s thesis, is to abstract Tam-

samani’s iteration process to obtain a theory of M -enriched categories,

weak in Segal’s sense, for a model category M .

7.1 Simplicial objects in M

The original definitions of Segal category, Tamsamani n-category, and

Pelissier’s enriched categories, took as basic object a functorA : ∆o →M .

The first condition is that the image A0 of [0] ∈ ∆ should be a “discrete

object”, that is the image of a set under the natural inclusion Set →M

which sends a set X to the colimit of ∗ over the discrete category cor-

responding to X . This version of the theory therefore requires, at least,

some axioms saying that the functor Set →M is fully faithful and

compatible with disjoint unions. Thus A0 may be viewed as a set and

the expression x ∈ A0 means that x is an element of the correspond-

ing set, equivalently x : ∗ → A0. The higher elements of the simplicial

object will be denoted Am/.

Then, the Segal category condition says that the Segal maps

Am/ → A1/ ×A0 · · · ×A0 A1/

are supposed to be weak equivalences.

The pair of structural maps (∂0, ∂1) : A1/ → A0 × A0 serves to

decompose

A1/ =∐

x,y∈A0

A(x, y)


104Weak enrichment over a cartesian model category: an introduction

where A(x, y) is the inverse image of (x, y) ∈ A0×A0, or more precisely

A(x, y) := A1/ ×A0×A0 ∗

with the right map of the fiber product being given by (x, y) : ∗ → A0×

A0. We can similarly decompose

Am/ =∐

(x0,...,xm)∈Am+10

A(x0, . . . , xm)

and the Segal condition may be expressed equivalently as saying that

A(x0, . . . , xm) → A(x0, x1)× · · · × A(xm−1, xm)

is a weak equivalence in M .

7.2 Diagrams over ∆X

Upon closer inspection, most of the arguments about M -Segal cate-

gories can really be phrased in terms of the objects A(x0, . . . , xm); and

in these terms, the Segal condition involves only a product rather than

a fiber product. So, it is natural and useful to consider the objects

A(x0, . . . , xm) as the primary objects of study rather than the Am/.

This economizes some hypotheses and arguments about discrete objects

and fiber products.

This point of view has been introduced by Lurie [155]. For any set

X , define the category ∆X whose objects are finite linearly ordered

sets decorated by elements of X , that is to say an object of ∆X is an

ordered set [m] ∈ ∆ plus a map of sets x· : [m] → X . This pair will be

denoted (x0, . . . , xm), that is it is an m+ 1-tuple of elements of X . The

morphisms in the category ∆X are the morphisms of ∆ which respect

the decoration, so for example the three standard morphisms [1] → [2]

yield morphisms of the form

(x0, x1) → (x0, x1, x2), (x1, x2) → (x0, x1, x2), (x0, x2) → (x0, x1, x2).

Now, an M -Segal category will be a pair (X,A) where X is a set, called

the set of objects, and A : ∆oX → M is a functor denoted by

([m], x·) = (x0, . . . , xm) 7→ A(x0, . . . , xm)

or just (x0, . . . , xm) 7→ A(x0, . . . , xm) if there is no danger of confusion,

such that the Segal maps

A(x0, . . . , xm) → A(x0, x1)× · · · × A(xm−1, xm)

7.3 Hypotheses on M 105

are weak equivalences in M . At m = 0 the Segal condition says that

A0(x0) → ∗ is a weak equivalence. This is a sort of weak unitality

condition, but for our purposes it is generally speaking better to impose

the strict unitality condition that A0(x0) = ∗ for any x0. This condition

becomes essential when we consider direct products.

At m = 1, the morphism space between two elements x, y ∈ X is

A(x, y). At m = 2 the usual diagram using the three standard mor-

phisms, serves to define the composition operation in a weak sense:

A(x, y)×A(y, z) ← A2(x, y, z) → A(x, z) (7.2.1)

with the leftward arrow being a weak equivalence in M . For higher

values of m we get the higher homotopy coherence conditions starting

with associativity at m = 3.

7.3 Hypotheses on M

In the weak enrichment, the composition operation is given by a diagram

of the form (7.2.1) above, using the usual direct product × in M and

where the leftward arrow is the Segal map which is required to be a weak

equivalence.

Therefore, the main condition which we need to impose upon M is

that it be cartesian, that is to say a monoidal model category whose

monoidal operation is the direct product. This insures that direct prod-

uct is compatible with the cofibrations and weak equivalences. Monoidal

model categories have been considered by many authors, see Hovey [120]

for example, and the cartesian theory is a special case. This condition

will be discussed in Chapter 10.

For convenience we also impose the conditions that M be left proper,

and tractable. Tractability is Barwick’s slight modification of J. Smith’s

notion of combinatorial model category. Recall that a combinatorial model

category is a cofibrantly generated one whose underlying category is lo-

cally presentable—locally presentable categories are the most appropri-

ate environment for using the small object argument, one of our staples.

Barwick’s tractability adds the condition that the domains of the gen-

erating cofibrations and trivial cofibrations, be themselves cofibrant ob-

jects. This is useful at some technical places in the small object argu-

ment. Our discussion of these topics is put together in Chapters 8 and

9.

In Section 12.7 we consider some additional hypotheses on M saying


that disjoint unions behave like we think they do; if M satisfies these

hypotheses then the discrete-set objects in M work well, and we can

use the notation Am/ for the disjoint union of the A(x0, . . . , xm). This

reduces to consideration of simplicial objects in M rather than functors

from ∆oX . If M is a category of presheaves over a connected category

Φ then it satisfies the additional hypotheses, and the category of M -

precategories discussed next will again be a category of presheaves (over

a quotient of ∆×Φ). The fact that iteratively we stay within the world

of presheaf categories, is convenient if one wants to think of the small

object argument in a simplified way.

7.4 Precategories

A tractable left proper cartesian model category M is fixed. For the

original case of n-categories, M would be the model category for (n−1)-

categories constructed according to the inductive hypothesis. In order

for the induction to work, the main goal is to construct from M a new

model category, whose objects represent up to homotopy the M -enriched

Segal categories, and which satisfies the same hypotheses of tractability,

left properness, and the cartesian condition.

If M satisfies the additional hypotheses on disjoint unions, then an

M -enriched category is a functorA : ∆o →M such that A0 is a discrete

set also called Ob(A), and such that the Segal maps

An → A1/ ×A0 · · · ×A0 A1/

are weak equivalences in M .

However, looking at the category of all such functors, the Segal con-

dition is not preserved by limits or colimits of diagrams. It would be

preserved by homotopy limits, but not even by homotopy colimits, and

indeed the problem of taking a homotopy colimit of diagrams and then

imposing the Segal condition is our main technical difficulty.

So, in order to obtain a model category structure, we have to relax

the Segal condition. This leads to the basic notion of M -precategory.

Our utilisation of the word “precategory” is similar to but not the same

as that of [122]. The reader may refer to the introduction of [193] for a

discussion of this notion in the original n-categorical context.

If M satisfies the additional condition about disjoint unions, then an

M -precategory may be defined as a functor A : ∆o →M such that A0

7.5 Unitality 107

is a discrete set, that is to say a disjoint sum of copies of the coinitial

object ∗ ∈M .

In the more general case, an M -enriched precategory is a pair (X,A)

where X is a set (often denoted by Ob(A)), and A : ∆oX →M is a

functor satisfying the unitality condition thatA(x0) = ∗ for any sequence

of length zero (i.e. having only a single element).

In either of these situations, the category PC(M ) of such diagrams

is closed under limits and colimits, and furthermore if M is locally pre-

sentable then PC(M ) is locally presentable too. The category PC(M )

will thus serve as a suitable substrate for our model structure. The fi-

brant objects of the model structure should additionally satisfy the Segal

conditions.

An additional benefit of the notation ∆X is that it allows us to break

down the argument into two pieces, a suggestion of Clark Barwick [14].

Indeed, we obtain two different categories, PC(X ;M ) and PC(M ).

The first consists of all M -precategories with a fixed set of objects X .

It is just the full subcategory of the diagram category Func(∆oX ,M )

consisting of diagrams satisfying the strict unitality condition A0(x0) =

∗. The study of PC(X ;M ), considered first, is therefore almost the same

as the study of the category of M -valued diagrams on a fixed category

∆oX .

The category PC(M ) is obtained by letting X vary, with a natural

definition of morphism (X,A) → (Y,B). Once everything is well under

way and the objects of PC(M ) become our main objects of study, then

we will drop the set X from the notation: an object of PC(M ) will be

denoted A and its set of objects by Ob(A), but with the same letter for

the functor A : ∆oOb(A) →M , in other words A denotes (Ob(A),A).

7.5 Unitality

The strict unitality condition says that A(x0) = ∗. The reason for impos-

ing this condition, aside from its convenience, is that it is needed to ob-

tain the cartesian condition on the model category of M -precategories.

Indeed, if we don’t impose the unitality condition, then the precate-

gories must be allowed to have A(x0) arbitrary, even those would be

forced to be contractible by the Segal condition of length 0. Product

with a non-unital precategory such that B(x0, . . . , xm) = ∅ for all se-

quences of objects, is not compatible with weak equivalences (as will be

discussed in Section 19.3.1).


Given the fact that the Eckmann-Hilton argument rules out a number

of different approaches to higher categories, as we have seen in Chapter 4

but also as in Cheng and Makkai’s remark in Penon’s original definition

[67], we should justify why our version of the unitality condition which

says that A(x0) = ∗ doesn’t also lead to an Eckmann-Hilton argument.

The point is that in the Segal-style definitions, the composition is not

a well-defined operation. So, even if there exist cells which are supposed

to be the “identities”, there is not a single well-defined composition with

the identity. The degeneracies provide 2-cells which say that, for an i-

morphism f , some possible composition of the form 1t(f) f or f 1s(f)will be equal to f , but these choices (which we call respectively the left

and right degeneracies) are not the only possible ones. The main step of

the Eckmann-Hilton argument going from

r r r

f ⇓

1 ⇓

1 ⇓

g ⇓

to

r r

1 f = f ⇓

g 1 = g ⇓

involves glueing the left degeneracy for f on top to the right degeneracy

for g on the bottom, generating a coproduct of cells which doesn’t fit into

any canonical global composition operation for the four 2-morphisms at

once. And similarly for the step involving vertical compositions. The

information on composition with units which comes from the unitality

condition and the degeneracies of ∆, is luckily not enough to make the

Eckmann-Hilton argument work. Because we’re close to the borderline

here, it is clear that some care should be taken to verify everything re-

lated to the unitality condition in the technical parts of our construction.

Unitality will therefore be considered n the context of more general

up-to-homotopy finite product theories, in Chapter 13.

7.6 Rectification of ∆X -diagrams 109

7.6 Rectification of ∆X-diagrams

The reader should now be asking the following question: wouldn’t it be

better to consider M as some kind of higher category, and to look at weak

functors ∆oX →M ? This would certainly seem like the most natural

thing to do. Unfortunately, this idea leads to “bootstrapping” problems

both philosophical as well as practical. On the philosophical level, the

really good version of M as a higher category, is to think of M as being

enriched over itself. We exploit this point of view starting from Section

22.5 where M considered as an M -enriched category is called Enr(M ).

However, if we are looking to define a notion of M -enriched category,

then we shouldn’t start with something which is itself an M -enriched

category. One can imagine getting around this problem by noting that

M , considered as a category enriched over itself, is actually strictly

associative; however for looking at functors to M we need to go to a

weaker model, and we end up basically having at least to pass through

the notion of strict functors ∆X →M . One could alternatively say that

instead of requiring that M be considered as an M -enriched category, we

could look at a slightly easier structure such as the Dwyer-Kan simplicial

localization associated to M . In this case, we would need a theory of

weak functors from ∆oX to a simplicial category. This theory has already

been done by Bergner [33] [34], so it would be possible to go that route.

However, it would seem to lead to many notational and mathematical

difficulties.

Luckily, we don’t need to worry about this issue. It is well-known that

any kind of weak functors from a usual 1-category, to a higher cate-

gory such as comes from a model category, can be rectified (or “stric-

tified”) to actual 1-functors. This became apparent as early as [107]

where Grothendieck pointed out that fibered categories are equivalent to

strictly cartesian fibered categories. Since then it has been well-known to

homotopy theorists working on diagram categories, and indeed the var-

ious model structures on the category of diagrams Func(∆oX ,M ) serve

to provide model categories whose corresponding higher categories, in

whatever sense one would like, are equivalent to the higher category of

weak functors. In the context of diagrams towards Segal categories, an

argument is given in [117].

Due to the philosophical bootstrapping problem mentioned above, I

don’t see any way of making the argument given in the previous para-

graph into anything other than the heuristic consideration that it is.


But, taking it as a basic principle, we shall stick to the notion of a usual

functor ∆oX →M as being the underlying object of study.

7.7 Enforcing the Segal condition

We relaxed the Segal condition in order to get a good locally presentable

category PC(M ) of M -precategories. The Segal condition should then

be built into the model structure, for example it is supposed to be sat-

isfied by the fibrant objects. This guides our construction of the model

structure: a fibrant replacement should impose or “force” the Segal con-

dition, and such a process in turn tells us how to define the notion of

weak equivalence.

To understand this, one should view an M -enriched precategory as

being a prescription for constructing an M -enriched category by a collec-

tion of “generators and relations”. The notion of precategory was made

necessary by the need for colimits, so one should think of a precategory

as being a colimit of smaller pieces. The associated M -enriched category

should then be seen as the homotopy colimit of the same pieces, in the

model category we are looking for. That is to say it is an object specified

by generators and relations. This is explained in some detail for the case

of 1-categories in Section 16.8.

The calculus of generators and relations is the process whereby A

may be replaced with Seg(A) which is, in a homotopical sense, the

minimal object satisfying the Segal conditions with a map A → Seg(A).

Another way of putting it is that we enforce the Segal conditions using

the small object argument. In order to find the model category, we should

define and investigate closely this process of generating an M -enriched

category.

The construction Seg(A) doesn’t in itself change the set of objects

of A, so we can look at it in the smaller category PC(X,M ). There,

it can be considered as a case of left Bousfield localization. This way of

breaking up the procedure was suggested by Barwick. Luckily, the left

Bousfield localization which occurs here has a particular form which we

call “direct”, in which the the weak equivalences may be characterized

explicitly, and we develop that theory with general notations in Chap-

ter 11. Going to a more general situation helps to clarify and simplify

notations at each stage; it isn’t clear that these discussions would have

significant other applications although that cannot be ruled out. Contin-

uing in this way, we discuss in Chapter 13 the application of direct left

7.7 Enforcing the Segal condition 111

Bousfield localization to algebraic theories in diagram categories. This

formalizes the idea of requiring certain direct product maps to be weak

equivalences, with the objective of applying it to the Segal maps. Here

we refer implicitly to the theories of sketches and algebraic theories.

Then, in Chapters 14 and 16 we apply the preceding general discus-

sions to the case of M -enriched precategories, and define the operation

A 7→ Seg(A) which to an M enriched precategory A associates an M -

enriched category, i.e. a precategory satisfying the Segal condition. As

a rough approximation the idea is to “force” the Segal condition in a

minimal way, an operation that can be accomplished using a series of

pushouts along standard cofibrations.

The passage from precategories to Segal M -categories is inspired by

the workings of the theory of simplicial presheaves as developped by

Joyal and Jardine [125] [123]. Whereas their ultimate objects of interest

were simplicial presheaves satisfying a descent condition, it was most

convenient to consider all simplicial presheaves and impose a model

structure such that the fibrant objects will satisfy descent. The Segal

condition is very close to a descent condition as has been remarked by

Berger [30].

As in [123], we are tempted to use the injective model structure for

diagrams, defining the cofibrations to be all maps of diagrams A → B

which induce cofibrations at each stage An → Bn. It turns out that

a slightly better alternative is to use a Reedy definition of cofibration,

see Chapter 15. If M is itself an injective model category then they

coincide. It can also be helpful to maintain a parallel projective model

structurewhere the cofibrations are generated by elementary cofibrations

as originally done by Bousfield. However, the projective structure is not

helpful at the iteration step: it will not generally give back a cartesian

model category.

Once we have a construction A 7→ Seg(A) which enforces the Se-

gal condition, a map A → B is said to be a “weak equivalence” if

Seg(A) → Seg(B) satisfies the usual conditions for being an equiva-

lence of enriched categories, essential surjectivity and full faithfulness.

A map of M -enriched categories f : A → B is fully faithful if, for

any two objects x, y ∈ Ob(A) the map A(x, y) → B(f(x), f(y)) is

a weak equivalence in M . Taking a homotopy class projection π0 :

M → Set gives a truncation operation τ≤1 from M -enriched cate-

gories to 1-categories, and we say that f : A → B is essentially sur-

jective if τ≤1(f) : τ≤1(A) → τ≤1(B) is an essentially surjective map of

1-categories, i.e. it is surjective on isomorphism classes. The isomorphism


classes of τ≤1(A) should be thought of as the “equivalence classes” of

objects of A. Putting these together, we say that a map f : A → B is an

equivalence of M -enriched categories if it is fully faithful and essentially

surjective.

Now, we say that a map f : A → B of M -enriched precategories, is a

weak equivalence if the corresponding map between the M -enriched cate-

gories obtained by generators and relations Seg(f) : Seg(A) → Seg(B)

is an equivalence in the above sense. With this definition and any one

of the classes of cofibrations briefly referred to above and considered in

detail in Chapter 15, the specification of the model structure is com-

pleted by defining the fibrations to be the morphisms satisfying right

lifting with respect to trivial cofibrations i.e. cofibrations which are weak

equivalences.

7.8 Products, intervals and the model structure

The introduction of M -precategories together with the operation Seg

allows us to define pushouts of weakly M -enriched categories: if A → B

and A → C are morphisms of weak M -enriched categories, then the

pushout of diagrams ∆o →M gives an M -precategory B ∪A C. The

associated pushout in the world of weakly M -enriched categories is sup-

posed to be Seg(B ∪A C). Proving that the collections of maps we have

defined above, really do define a closed model category, may be viewed as

showing that this pushout operation behaves well. As came out pretty

clearly in Jardine’s construction [123] but was formalized in Smith’s

recognition principle [27] [84] [16], the key step is to prove that pushout

by a trivial cofibration is again a trivial cofibration.

Before getting to the proof of this property, one has to calculate some-

thing somewhere, which is what is done in the Chapters 18 and 19 lead-

ing up to the theorem that the calculus of generators and relations is

compatible with direct products:

Seg(A)× Seg(B) → Seg(A× B)

is a weak equivalence. Our proof of this compatibility really starts in

Chapter 18 about free ordered M -enriched categories. These may be

used as basic building blocks for the generators defining the model struc-

ture, so it suffices to check the product condition on them, which is then

done in Chapter 19.

The compatibility between Seg and direct products leads to what will

7.8 Products, intervals and the model structure 113

be the main part of the cartesian property for the model category which

is being constructed. This is a categorical analogue of the Eilenberg-

Zilber theorem for simplicial sets. It wouldn’t be true if we hadn’t kept

the degeneracy maps in ∆, and the strict unitality condition seems to

be essential too.

From this result on direct products, a trick lets us conclude the main

result for constructing the model structure via a Smith-type recognition

theorem: that trivial cofibrations are preserved by pushout. For that

trick, one requires also a good notion of interval, which was the subject

of Pelissier’s correction [171] to an errof in [193]. Although Pelissier dis-

cussed only the case of Segal categories enriched over the model category

K of simplicial sets, his construction transfers to PC(M ) by functo-

riality using a functor K →M . A somewhat similar correction was

made by Bergner in her construction of the model category structure for

simplicial categories originally suggested by Dwyer and Kan [33].

The construction of a natural “interval category” is described in Chap-

ter 20. It is a sort of versal replacement for the simple category 0 ←→ 1

with two isomorphic objects. This is the point where Pelissier’s cor-

rection [171] of [193] comes in, and in order to make the process fully

iterative we just have to point out that an interval for the case of the

standard model category M = K of simplicial sets, leads by functo-

riality to an interval for any other M . The good version of the versal

interval constructed by Pelissier [171] is similar to the interval object

for dg-categories subsequently introduced by Drinfeld [83]. We modify

slightly Pelissier’s construction, but one could use his original one too.

Once all of these ingredients are in place, we can construct the model

structure in Chapter 21.We obtain a model category structure onPC(M )

which again satisfies all of the hypotheses which were required of M , so

the process can be iterated.

Starting with the trivial model category structure on Set, the n-th

iterate PCn(Set) is the model category structure for n-precategories as

considered in [193]. If instead we start with the Kan-Quillen model cate-

gory K of simplicial sets, then PCn(K ) is the model category of Segal

n-precategories which was used in the work [117] about n-stacks. We

discuss these iterations for weak n-categories (which are Tamsamani’s

n-nerves) and Segal n-categories, together with a few variants where the

initailzing category of sets is replaced by a category of graphs or other

things, in Chapter 22.

The internal Hom operation then leads to a category enriched over


our new model category, which in the iterative scheme for n-categories

gives a construction of the n+ 1-category nCAT .

The last part of the book, not yet included in the present version, will

be dedicated to considering how to write in this language some basic

elements of higher category theory, such as inverting morphisms, and

limits and colimits. We also hope to discuss the Breen-Baez-Dolan sta-

bilization hypothesis, about the behavior of the theories of n-categories

for different values of n.

PART II

CATEGORICAL PRELIMINARIES

8

Some category theory

In this chapter, we regroup various things which can be said in the

context of abstract category theory. Our discussion is based in large

part on the book of Adamek and Rosicky [2] about locally presentable

and accessible categories. Refer there for historical remarks about these

notions. The applicability of this theory to model categories came out

with J. Smith’s notion of combinatorial model category [198], slightly

modified by Barwick with his notion of tractable model category [16].

One of our main goals is to provide a fairly general discussion of

the notion of cell complex in a locally presentable category. Hirschhorn

has formalized the use of cell complexes for the small object argument

and left Bousfield localization, in [116]. However, he used an additional

assumption of a monomorphism property of elements of the generating

set of arrows I ⊂ Arr(M ), encoded in his notion of cellular model

category. We would like to avoid this hypothesis. Indeed, one of the

main examples which we can use to start out our induction is the model

category on Set, but as Hirschhorn pointed out this is not cellular. It

turns out that a somewhat more abstract approach to cell complexes

works pretty well.

Our discussion covers much the same materiel as Lurie in the appendix

to [153]. Lurie introduces a notion of “tree” generalizing the standard

transfinite cell-addition process. The basic idea is that once we have

attached a certain number of cells, the next cell is attached along a

κ-presentable subcomplex, but this information is lost under the usual

indexation by an ordinal. In our discussion, just to be different, we’ll

stick to the standard ordinal presentation, but we introduce a category

of “inclusions of cell complexes”, and show that the category of κ-small

inclusions of cell complexes into a given one, is κ-filtered. Roughly speak-

ing, an inclusion of cell complexes corresponds to a downward-closed


118 Some category theory

subset of a tree. We sketch a proof of Lurie’s theorem [153, Proposition

A.1.5.12] that cofibrations are cell complexes over κ-small cofibrations,

rather than just retracts of such.

The main application of this result is to construct the generating set

for injective cofibrations. Again we give a brief account of a proof of the

main technical result in the present chapter, although the reader can

also refer to [153] and [16].

This discussion prepares the way for the “recognition principle” intro-

duced in Chapter 9, based on Smith’s recognition principle as reported

by Barwick [16]. Our addition is to give a statement which encodes the

accessibility argument. The advantage is that the notion of accessibility

no longer appears in the statement, so we can then use that in later

chapters to construct model categories without needing to discuss the

notion of accessibility anymore.

So, in a certain sense what we are doing here is to evacuate some of

the more technical details in the theory of model categories, towards

these first two chapters. We hope that this will be helpful to the reader

who wishes to avoid this kind of discussion: if willing to take for granted

the recognition principle which will be stated as Theorem 9.9.7 in the

next chapter, the reader may largely skip over the most technical parts

of these first two chapters.

In order to avoid repetitive language, we often apply the following

conventions about universes. We assume given at least two universes

U ∈ V. Recall that these are sets which themselves provide models for

ZFC set theory. A category will mean a category object in V. An example

is the category SetU of sets in U. A small category will be a category

object in U, which is also one in V. Often a category C will have small

morphism sets, that is for any x, y ∈ Ob(C ) the set HomC (x, y) ∈ V is

isomorphic to a set in U.

Depending on context, the word “category” can sometimes mean “small

category”, or sometimes “category with small morphism sets”. However,

when we need to consider categories outside of V this will be explicitly

mentioned.

Recall that an ordinal is a set a, such that if x ∈ y and y ∈ a then

x ∈ a; and such that a is well-ordered by the strict relation x < y ⇔ x ∈

y for x, y ∈ a. For the corresponding non-strict order relation we then

have x ≤ y ⇔ x ⊂ y, and for any x ∈ a the successor of x is x ∪ x.

A cardinal is an ordinal a with the property that for any b ∈ a, b is

not isomorphic to a. Any set x has a unique cardinality |x| which is a

cardinal such that x ∼= |x|.


An ordinal (resp. cardinal) of U is an ordinal (resp. cardinal) which is

an element of U. These are the ordinals (resp. cardinals) for the model

of set theory given by U. In particular, for any x ∈ U we have |x| ∈ U.

We say that an ordinal α is approached by a sequence of cardinality λ if

there is a subset x ⊂ α with |x| = λ, such that α is the least upper bound

of x. A cardinal κ is regular if it is not approached by any sequence of

cardinality < κ.

8.1 Locally presentable categories

Fix a regular cardinal κ. A category Φ is said to be κ-filtered if for any

collection of< κ objectsXi ∈ C , there exists an object Y and morphisms

Xi → Y ; and for any pair of objects X and Y , and any collection of

< κ morphisms fi : X → Y there exists a morphism g : Y → Z such

that all the gfi are equal. Note that taking an empty set of objects in

the first condition implies that C is nonempty. A κ-filtered colimit is a

colimit over a κ-filtered index category. See [2, Remark 1.21].

Let C be a category. We assume that C admits κ-filtered colimits.

Then, say that an object X ∈ C is κ-presentable if, for any κ-filtered

colimit colimi∈ΦYi = Z, the map

colimi∈ΦHomC (X,Yi) → HomC (X,Z)

is an isomorphism of sets. An object Z ∈ C is said to be κ-accessible if

it can be expressed as a κ-filtered colimit of κ-presentable objects.

Definition 8.1.1 Let κ be a regular cardinal. A category C with small

morphism sets is called κ-accessible if:

(1)—C admits κ-filtered colimits;

(2)—the full subcategory of κ-presentable objects is equivalent to a small

category;

(3)—every object of C is κ-accessible.

Furthermore, if C admits all small colimits then we say that C is locally

κ-presentable. A category which is locally κ-presentable for some regular

cardinal κ is called locally presentable.

For the countable cardinal κ = ω, the terminology “locally finitely

presentable” is interchangeable with “locally ω-presentable”.

Theorem 8.1.2 Suppose C is locally presentable. Then it is complete,

i.e. it admits small limits too. Each object has only a small set of sub-

objects up to isomorphism. All κ-filtered colimits commute with κ-small


limits (i.e. limits over categories cardinality < κ). For any X ∈ C , the

subcategory Cκ/X of κ-presentable objects of C , is κ-filtered and X is

canonically the colimit of the forgetful functor on Cκ/X.

If C is locally κ-presentable then for any regular cardinal κ > κ it is

locally κ′-presentable too.

Proof See [159], or [2], Proposition 1.22, Corollary 1.28, Remark 1.56

and Proposition 1.59. For the last sentence see [2, page 22].

Lemma 8.1.3 Suppose Ψ is a small category, and C is locally κ-

presentable. Then the category Func(Ψ,C ) of diagrams from Ψ to C ,

is locally κ-presentable. The κ-presentable diagrams in Func(Ψ,C ) are

exactly the functors F : Φ → C such that F (a) is κ-presentable in C

for every a ∈ Φ.

Proof See Makkai and Pare [159], or Adamek and Rosicky [2, Corollary

1.54].

Corollary 8.1.4 Suppose Ψ is a small category, then the category

Presh(Ψ) of presheaves of sets on Ψ, is locally finitely presentable.

Proof The category of sets is locally finitely presentable.

If C is a category, let Arr(C ) be the category of arrows of C , whose

objects are the diagrams of shape Xf→ Y in C . The morphisms in

Arr(C ) from Xf→ Y to X ′ f ′

→ Y ′ are the commutative squares

X → X ′

Y

f

↓→ Y ′

f ′

↓

.

Corollary 8.1.5 Suppose C is a locally κ-presentable category. Then

Arr(C ) is locally κ-presentable.

Proof Indeed, Arr(C ) = Func(E ,C ) where E is the category with

two objects 0, 1 and a single morphism 0 → 1 besides the identities.

Therefore 8.1.3 applies.

In a similar way, any category of commutative diagrams of a given

shape in a locally presentable category, will again be locally presentable.

A functor q : α → β is cofinal if:

—for any object i ∈ β there exists j ∈ α and an arrow i → q(j);


—for any pair of arrows i → q(j) and i → q(j′) in β there are arrows

j → j′′ and j′ → j′′ in α such that the diagram

i → q(j)

j′↓

→ q(j′′)

↓

commutes (see [2, 0.11]).

Recall that if q : α → β is a cofinal functor, then it induces an

equivalence between the theory of colimits indexed by β and the theory

of colimits indexed by β, see [2, page 4].

A basic and motivating example of a locally presentable category is

when Φo is a site, and M ⊂ SetΦU is the subcategory of sheaves. In

this case H∗ just denotes the identity inclusion, whereas T = H! is the

sheafification functor. This motivates the following characterization.

Proposition 8.1.6 A V-category C with U-small morphism sets is

locally presentable if it has U-small limits and colimits, and if there exists

a U-small category Φ and adjunction

H! : SetΦU ←→ C : H∗

and a regular cardinal κ ∈ U such that H!H∗ is the identity, and the

composition T := H∗H! : SetΦU→ Set

ΦUcommutes with β-directed col-

imits. Or equivalently, that H∗ itself preserves κ-directed colimits.

Proof This rephrases [2], Theorem 1.46, using an adjoint pair of func-

tors such that one composition is the identity, instead of a full reflective

subcategory.

A locally κ-presentable category C will in general have the property

that |Hom(X,Y )| > κ for two κ-presentable objects. For example, SetΦU

is locally κ-presentable whatever the size of Φ, but if |Φ| > κ then there

can be > κmorphisms between κ-presentable objects. This is rectified by

taking κ big enough, but the bound has to be exponential in κ because

we look at maps from objects of size < κ.

Corollary 8.1.7 Suppose C is a locally presentable category. There

is a regular cardinal κ such that C is locally κ-presentable, such that

for any regular cardinals λ, µ > κ and objects X,Y such that X is λ-

presentable and Y is µ-presentable, the set of morphisms HomC (X,Y )

has size < µλ.


Proof Suppose C is κ0-presentable to begin with. The total cardinal-

ity of a presheaf A ∈ SetΦU is the sum of the cardinalities of the val-

ues A(x) for x ∈ Φ. Use the characterisation of the previous propo-

sition, and choose a new regular cardinal κ1 > sup(κ0, |Φ|). For any

λ ≥ κ1 the λ-presentable objects of C are those of the form H!(A)

for presheaves A : Φ → SetU of total cardinality < λ (see [Example

1.31]AdamekRosicky). Choose a regular cardinal κ ≥ κ1 so that the to-

tal cardinality of H∗H!(B) is < κ for any presheaf B of total cardinality

< κ1.

Now suppose λ, µ > κ. By [2, Remark 1.30(2)], the λ-presentable

objects of C are κ1-filtered colimits of size < λ, of κ1-presentable ob-

jects (and the same for µ). Suppose X and Y are λ-presentable and

µ-presentable objects respectively. Write X = colimi∈IH!(Ai) (resp.

Y = colimj∈JH!(Bj)) where I (resp. J) is a κ1-filtered category of size

< λ (resp. of size < µ), and Ai and Bj are presheaves of total cardinality

< κ1. Then H∗H!(Bj) has total cardinality < κ. Now

HomC (X,Y ) = limi∈I

(colimj∈JHomC (Ai, H∗H!(Bj))) .

This has size < µλ.

One should also be able to prove this using the characterization of

locally presentable categories as categories of models of limit theories [2,

Theorem 5.30].

The following lemma will be useful in dealing with unitality conditions

in Chapter 13.

Lemma 8.1.8 Suppose M is a category with coinitial object ∗, and

suppose α is a nonempty connected small category (that is, a category

whose nerve is a connected simplicial set). Then the colimit of the con-

stant functor C· : α →M defined by Ci = ∗ for all i ∈ α, exists and is

equal to ∗.

Proof There is a unique compatible system of morphisms φi : Ci =

∗ → ∗. We claim that this makes ∗ into a colimit of C·. Suppose U ∈M

and ψi : Ci → U is a compatible system of morphisms. Pick i0 ∈ α

and use f := ψi0 : Ci0 = ∗ → U . We claim that for any j ∈ α the

composition fφj is equal to ψj . Let α′ be the subset of objects of α

for which this is true, nonempty since it contains i0. If j′ ∈ α′ and if

g : j → j′ is an arrow of α then

fφj = fφj′Cg = ψj′Cg = ψj


so j ∈ α′. Suppose j′ ∈ α′ and if g : j′ → j is an arrow of α. Then

fφjCg = fφj′ = ψj′ = ψjCg

but Cg is an isomorphism so fφj = ψj and j ∈ α′. These two steps imply

inductively, using connectedness of α, that α′ = Ob(α). Thus fφ· = ψ·.

Clearly f is unique, so we get the required universal property.

8.2 Monadic projection

Suppose C is a category, and R ⊂ C a full subcategory. We assume that

R is stable under isomorphisms.

A monadic projection from C to R is a functor F : C → C together

with a natural transformation ηX : X → F (X), such that:

(Pr1)—F (X) ∈ R for all X ∈ C ;

(Pr2)—for any X ∈ R, ηX is an isomorphism; and

(Pr3)—for any X ∈ C , the map F (ηX) : F (X) → F (F (X)) is an

isomorphism.

Lemma 8.2.1 Suppose (F, η) is a monadic projection. Then the two

isomorphisms F (ηX) and ηF (X) from F (X) to F (F (X)) are equal.

Proof Naturality of η with respect to the morphism ηX gives the com-

mutative diagram

XηX→ F (X)

F (X)

ηX

↓F (ηX)

→ F (F (X))

ηF (X)

↓

.

For any X ∈ R, ηX is an isomorphism so composing with its inverse we

get F (ηX) = ηF (X).

On the other hand, we can also apply F to the above diagram. By

(Pr3), for any X ∈ C we have that F (ηX) is an isomorphism so again

composing with its inverse we conclude that F (F (ηX)) = F (ηF (X)) for

any X ∈ C .


For arbitrary X ∈ C , consider the diagram

F (X)F (ηX)

→ F (F (X))

F (F (X))

ηF (X)

↓F (F (ηX))

→ F (F (F (X)))

ηF (F (X))

↓

.

It commutes by naturality of η with respect to the morphism F (ηX). On

the other hand, by the first statement we proved above, and noting that

F (X) ∈ R by (Pr1), we have ηF (F (X)) = F (ηF (X)). On the other hand,

by the second statement we proved above, F (F (ηX)) = F (ηF (X)). We

have now shown that both second maps in the two equal compositions

of this diagram, are the same isomorphism. It follows that the two first

maps along the top and the left side, are the same. This proves the

lemma.

Proposition 8.2.2 Suppose R ⊂ C are as above, and (F, η) and (G,ϕ)

are two monadic projections from C to R. Then, for any X ∈ M the

maps F (ϕX) : F (X) → F (G(X)) and G(ηX) : G(X) → G(F (X)) are

isomorphisms. The diagram of isomorphisms

F (X)F (ϕX)

→ F (G(X))

G(F (X))

ϕF (X)

↓

←G(ηX)

G(X)

ηG(X)

↑

commutes.

Proof Define the functor H(X) := G(F (X)), with a natural transfor-

mation ψX : X → H(X) defined as the composition

XϕX→ G(X)

G(ηX )→ G(F (X)).

Naturality of the transformation ϕ with respect to the morphism ηX


gives a commutative diagram

XϕX→ G(X)

F (X)

ηX

↓ϕF (X)→ G(F (X))

G(ηX)

↓

giving the the alternative expression for ψ,

ψX = G(ηX)ϕX = ϕF (X)ηX . (8.2.1)

We claim that (H,ψ) is again a monadic projection from C to R. The

first (Pr1) is a direct consequence of the same conditions for F and G.

For the second condition, suppose X ∈ R. Then ηX is an isomorphism

by (Pr2) for (F, η), and ϕF (X) is an isomorphism by (Pr2) for (G,ϕ)

plus (Pr1) for F . By the expression (8.2.1) we get that ψX = ϕF (X)ηXis an isomorphism which is (Pr2) for (H,ψ). One could instead use the

expression ψX = G(ηX)ϕX and the fact that a functor G preserves

isomorphisms.

For the third condition, note from (8.2.1) again, that H(ψX) is ob-

tained by applying G to the composed map

F (X)F (ηX)→ F (F (X))

F (ϕF (X))→ F (G(F (X))). (8.2.2)

The first arrow F (ηX) is an isomorphism by (Pr3) for (F, η). Consider

the diagram

F (X)ηF (X)

→ F (F (X))

G(F (X))

ϕF (X)

↓ηG(F (X))

→ F (G(F (X)))

F (ϕF (X))

↓

.

It commutes by naturality of η with respect to the morphism ϕF (X). The

right vertical arrow is the the second arrow in the composition (8.2.2).

The top arrow is an isomorphism by (Pr1) and (Pr2) for (F, η). The

left arrow is an isomorphism by (Pr1) for F and (Pr2) for (G,ϕ). The

bottom arrow is an isomorphism by (Pr1) for G and (Pr2) for (F, η). It

follows that the right vertical arrow F (ϕF (X)) is an isomorphism.


Now apply the above conclusions together with the fact that G pre-

serves isomorphisms, in the expression

H(ψX) = G(F (ϕF (X)) F (ηX)

)

to conclude that H(ψX) is an isomorphism. This completes the proof of

(Pr3) to show that (H,ψ) is a monadic projection.

Continue now the proof of the proposition. We have a morphism

G(X)G(ηX)→ G(F (X)) = H(X).

Consider the diagram

G(X)G(ϕX)

→ G(G(X))

H(X)

G(ηX)

↓H(ϕX)

→ H(G(X))

G(ηG(X))

↓H(G(ηX))

→ H(H(X))

.

The square commutes because it is obtained by applying G to the nat-

urality diagram for η with respect to the morphism ϕX . The top arrow

is an isomorphism by (Pr3) for (G,ϕ). The middle vertical arrow is an

isomorphism by applying G to (Pr2) for (F, η) and using (Pr1) for G.

This shows that the composition in the square is an isomorphism, so

we deduce that the composition of the left bottom arrow with the left

vertical arrow is an isomorphism.

The composition along the bottom is equal to H(ψX), indeed ψX =

G(ηX)ϕX by definition. Hence, by (Pr3) for (H,ψ) which was proven

above, the composition along the bottom is an isomorphism.

The morphism H(ϕX) at the bottom of the square, now has maps

which compose on the left and the right to isomorphisms. It follows that

this map is an isomorphism, and in turn that the left vertical arrow

G(ηX) is an isomorphism. This is one of the statements to be proven in

the proposition.

The other statement, that F (ϕX) is an isomorphism, is obtained by

symmetry with the roles of F and G reversed.

To finish the proof, we have to show that the square diagram of isomor-

phisms commutes (which means that it commutes as a usually shaped

diagram when the inverses of the isomorphisms are included).

Apply FG to the diagram in question, and add on another square to


get the diagram of isomorphisms

FGF (X)FGF (ϕX)

→ FGFG(X)

FGGF (X)

FG(ϕF (X))

↓

←FGG(ηX)

FGG(X)

FG(ηG(X))

↑

FGF (X))

FG(ϕF (X))

↑

←FG(ηX)

FG(X).

FG(ϕX)

↑

The bottom square commutes by FG applied to the naturality square

for ϕ with respect to ηX . On the left side, we have the same isomorphism

going in both directions. Consider the outer square

FGF (X)FGF (ϕX)

→ FGFG(X))

FGF (X))

wwwwwwwww←FG(ηX)

FG(X).

FG(ηG(X)ϕX)

↑

It is obtained by applying FG to the square

XηX→ F (X)

G(X)

ϕX

↓ηG(X)→ FG(X)

F (ϕX)

↓

which commutes by naturality of η with respect to ϕX . Since the outer

square, and the bottom square of the above diagram of isomorphisms

commute, it follows that the upper square commutes. The upper square

was obtained by applying FG to the square of isomorphisms in R in

question, and FG is a functor which is naturally isomorphic to the iden-

tity functor on R. Therefore the diagram of the proposition, which is a

diagram of isomorphisms in R, commutes.


8.3 Miscellany about limits and colimits

Here is an elementary observation about limits.

Lemma 8.3.1 Suppose F : I → J is a functor between small cate-

gories, suppose M is cocomplete, and suppose B : J →M is a diagram.

Pullback along F induces the diagram B F which is noted F ∗B. There

is an induced map in M

colimIF∗B → colimJB

Proof Let CI and CJ be the “constant diagram” constructions. One

way of defining colimits is by adjunction with C·. We have a universal

map in M J

B → CJ(colimJB),

and applying restriction along F we get a map

F ∗B → F ∗(CJ (colimJB)) = CI(colimJB).

By universality of the map B F → CI(colimIF∗B) we get a factoriza-

tion

F ∗B → CI(colimI(B F )) → CI(colimJB)

for a unique map colimIF∗B → colimJB which is the map in question

for the lemma.

Here is another fact useful in the construction of adjoints.

Lemma 8.3.2 Suppose p : α → β is a functor between small cate-

gories, and suppose A : β →M is a diagram with values in a cocom-

plete category M . Then there is a natural map colimαp∗(A) → colimβA,

satisfying compatibility conditions in case of compositions of functors.

Proof Indeed, there is a tautological natural transformation of β-diagrams

from A to the constant diagram with values colimβA, and the pull-

back of this natural transformation to α is a natural transformation of

α-diagrams from p∗(A) to the constant diagram with values colimβA,

which gives the map colimαp∗(A) → colimβA in question. If q : δ → α

is another functor then the composition of the maps for p and for q

colimαq∗(p∗(A)) → colimαp

∗(A) → colimβA

is the natural map for pq. Similarly, if p is the identity functor then the

associated natural map is the identity.

8.4 Diagram categories 129

8.4 Diagram categories

Suppose Φ is a small category and M a category. Consider the diagram

category Func(Φ,M ) of functors Φ →M . If M is complete (resp. co-

complete) then so is Func(Φ,M ) and limits (resp. colimits) of diagrams

are computed objectwise, that is over each object of Φ.

Suppose f : Φ → Ψ is a functor between small categories. Given

any diagram A : Ψ →M then the composition A f is a diagram

Φ →M which will also be denoted f∗(A). This gives a functor f∗ :

Func(Ψ,M ) → Func(Φ,M ). If M is complete (resp. cocomplete) then

the functor f∗ preserves limits (resp. colimits) since they are computed

objectwise in both Func(Φ,M ) and Func(Ψ,M ).

We consider the left and right adjoints of f∗. This parallels the similar

discussion worked out with A. Hirschowitz in in [117, Chapter 4] but of

course these things are of a nature to have been well-known much earlier.

See also [153, A.2.8.7].

If M is cocomplete then we can construct a left adjoint

f! : Func(Φ,M ) → Func(Ψ,M )

as follows. For any object y ∈ Ψ consider the category f/y of pairs (x, a)

where x ∈ Φ and a : f(x) → y is an arrow in Ψ. There is a forgetful

functor rf,y : f/y → Φ sending (x, a) to x.

Suppose A ∈ Func(Φ,M ). Put f!(A)(y) := colimf/yr∗f,y(A). Suppose

g : y → y′ is an arrow. Then we obtain a functor cg : f/y → f/y′

sending (x, a) to (x, ga). Furthermore this commutes with the forgetful

functors in the sense that rf,y′ cg = rf,y. Thus r∗f,y(A) = c∗g(r

∗f,y′ (A)).

Applying the above remark about colimits, we get a natural map

f!(A)(y) := colimf/yc∗g(r

∗f,y′(A)) → colimf/y′r

∗f,y′(A) =: f!(A)(y

′).

Using the last part of the paragraph about colimits above, we see that

this collection of maps turns f!(A) into a functor from Ψ to M , that

is an object in Func(Ψ,M ). The construction is functorial in A so it

defines a functor

f! : Func(Φ,M ) → Func(Ψ,M ).

The structural maps for the colimit defining f!(A)(y) are mapsA(x) → f!(A)(y)

for any a : f(x) → y. In particular when y = f(x) and a is the iden-

tity we get maps A(x) → f!(A)(f(x)) = f∗f!(A)(x). This is a natural

transformation from the identity on Func(Φ,M ) to f∗f!.

On the other hand, suppose A = f∗(B) for B ∈ Func(Ψ,M ). Then,


for any (x, a) ∈ f/y we get a map r∗f,y(f∗(B))(x, a) = B(f(x))

B(a)→ B(y).

This gives a map from r∗f,y(f∗(B)) to the constant diagram with values

B(y), hence a map on the colimit

f!f∗(B)(y)colimf/yr

∗f,y(f

∗(B)) → B(y).

It is functorial in y and B so it gives a natural transformation from f!f∗

to the identity.

Lemma 8.4.1 Supposing that M is cocomplete and with these natural

transformations, f! becomes left adjoint to f∗ and one has the formula

f!(A)(y) = colimf/yr∗f,y(A).

Proof SupposeA ∈ Func(Φ,M ) andB ∈ Func(Ψ,M ). A mapA → f∗(B)

consists of giving, for each x ∈ Φ, a map A(x) → B(f(x)). This is equiv-

alent to giving, for each y ∈ Ψ and (x, a) ∈ f/y, a map A(x) → B(y)

subject to some naturality constraints as x, a, y vary. This in turn is

the same as giving a map of f/y-diagrams from r∗f,y(A) to the constant

diagram with values B(y), which in turn is the same as giving a map

from f!(A) = colimf/yr∗f,y(A) to B. It is left to the reader to verify that

these identifications are the same as the ones given by the above-defined

adjunction maps.

Lemma 8.4.2 Suppose that M is complete. Then f∗ has a right ad-

joint denoted f∗ given by the formula

f∗(A)(y) = limf\ys∗f,y(A)

where f\y is the category of pairs (z, u) where z ∈ Φ and f(z)u→ y is

an arrow in Ψ, and sf,y : f\y → Φ is the forgetful functor.

Proof Apply the previous lemma to the functor fo : Φo → Ψo for

diagrams in the opposite category M o.

8.5 Enriched categories

Enriched categories have been familiar objects for quite a while, see

Kelly’s book [139]. Our overall goal is to discuss a homotopical analogue

susceptible of being iterated, so it is worthwhile to recall the classical

theory. Furthermore, this will provide an important intermediate step

of our argument: a weakly enriched category over a model category M

gives rise to a ho(M )-enriched category in the classical sense, and this


construction is conservative for weak equivalences. This is used notably

for Proposition 14.6.4.

Suppose E is a category admitting finite direct products. This includes

existence of the coinitial object ∗ which is the empty direct product.

If X is a set, then a E -enriched category on object set X is a collection

of objects A(x, y) ∈ E for x, y ∈ X , together with morphisms A(x, y)×

A(y, z) → A(x, z) and ∗ → A(x, x), such that for any x, y the composed

map

∗ ×A(x, y) → A(x, x) ×A(x, y) → A(x, y)

is the identity; the composed map

A(x, y)× ∗ → A(x, y) ×A(y, y) → A(x, y)

is the identity; and for any x, y, z, w the diagram

A(x, y)×A(y, z)×A(z, w) → A(x, y) ×A(y, w)

A(x, z)×A(z, w)

↓

→ A(x,w)

↓

commutes.

An E -enriched category is a pair (X,A) as above, but often this will

be denoted just by A with X = Ob(A). A functor between E -enriched

categories f : A → B consists of a map of sets f : Ob(A) → Ob(B),

and for each x, y ∈ Ob(A) a morphism fx,y : A(x, y) → B(f(x), f(y))

in E , such that the diagrams

∗ → A(x, x)

∗↓

→ B(f(x), f(x))

↓

andA(x, y)×A(y, z) → A(x, z)

B(f(x), f(y))×B(f(y), f(z))

↓

→ B(f(x), f(z))

↓

commute.


Let Cat(E ) denote the category of E -enriched categories. It admits

direct products too: if A and B are E -enriched categories the

Ob(A×B) = Ob(A)×Ob(B),

and for (x, x′) and (y, y′) in Ob(A)×Ob(B) we have

(A×B)((x, x′), (y, y′)) = A(x, y) ×B(x′, y′).

Hence this construction can be iterated and we can obtain Catn(E ), the

category of strict n-categories enriched in E at the top level.

Starting with E = Set yields Cat(Set) = Cat, and Catn(Set) is

the category of strictly associative and strictly unital n-categories. These

objects have been studied a great deal. However, as we have seen in

Chapter 4, these objects do not have a sufficiently rich homotopy theory,

in particular the groupoid objects therein do not model homotopy types

in any reasonable way. This obervation is the motivation for considering

weakly associative objects as in the remainder of this work.

If ϕ : E → E ′ is a functor compatible with direct products, apply-

ing it to the morphism objects of an E -enriched category A gives an

E ′-enriched category Cat(ϕ)(A) with the same set of objects, and mor-

phism objects defined by

Cat(ϕ)(A)(x, y) := ϕ(A(x, y)).

We apply this in particular to the functor τ≤0 : E → Set defined by

τ≤0(E) := HomE (∗, E). Define τ≤1 := Cat(τ≤0), i.e.

τ≤1A(x, y) = HomE (∗, A(x, y)).

With this, τ≤1A ∈ Cat(Set) is a usual category. Let Isoτ≤1A denote its

set of isomorphism classes. A functor A → B of E -enriched categories

is said to be essentially surjective if the induced map

Isoτ≤1A → Isoτ≤1B

is surjective. A functor f : A → B of E -enriched categories is said to be

fully faithful if, for each x, y ∈ Ob(A) the morphism

fx,y : A(x, y) → B(f(x), f(y))

is an isomorphism in E . A functor is an equivalence of E -enriched cate-

gories if it is essentially surjective and fully faithful.

These definitions are useful because, as we shall see in Chapter 14, a

morphism of M -enriched precategories is a global weak equivalence if

and only if the associated morphism of ho(M )-enriched categories is an


equivalence in the present sense. So, we can already obtain versions of

some of the main closure properties: closure under retracts and 3 for 2.

Lemma 8.5.1 Suppose ϕ : E → E ′ is a functor commuting with

direct products. Suppose f : A → B is an equivalence of E -enriched

precategories. Then

Cat(ϕ)(f) : Cat(ϕ)(A) → Cat(ϕ)(B)

is an equivalence of E ′-enriched precategories. This applies in particular

to the functor ϕ = τ≤0, to conclude that τ≤1(f) : τ≤1(A) → τ≤1(B)

is an equivalence of categories, hence f induces an isomorphism of sets

Isoτ≤1A ∼= Isoτ≤1B.

Proof The functor ϕ sends ∗E to ∗E ′ so it induces a map

HomE (∗, A) → HomE ′(∗, ϕ(A)).

This natural transformation induces a natural transformation of functors

τ≤1,ϕ : τ≤1,E → τ≤1,E ′ Cat(ϕ)

from Cat(E ) to Cat, which is the identity on underlying sets of objects.

Therefore the resulting natural transformation

Isoτ≤1,ϕ(A) : Isoτ≤1,E (A) → Isoτ≤1,E ′(Cat(ϕ)A)

is surjective. It follows that if f : A → B is essentially surjective in

Cat(E ) then Cat(ϕ)f : Cat(ϕ)A → Cat(ϕ)B is also essentially sur-

jective. On the other hand, ϕ takes isomorphisms to isomorphisms, so if

f is fully faithful then for any x, y ∈ Ob(Cat(ϕ)A) = Ob(A), the map

Cat(ϕ)(f)x,y = ϕ(fx,y) : ϕ(A(x, y)) → ϕ(B(f(x), f(y)))

is an isomorphism in E ′. This shows that if f is an equivalence in Cat(E )

then Cat(ϕ)(f) is an equivalence in Cat(ϕ)(E ′).

For the second part of the statement, apply this to ϕ := τ≤0 which

preserves products.

Lemma 8.5.2 In any category E , the class of isomorphisms is closed

under retracts and satisfies 3 for 2.

Proof It is easy to see for the category of sets. A retract of objects

in E gives a retract of representable functors to sets; so if we have a

retract of an isomorphism then the resulting retracted natural transfor-

mation is a natural isomorphism, and a morphism in E which induces

an isomorphism between representable functors, is an isomorphism.


The 3 for 2 property is easy to see using the inverses of the isomor-

phisms in question.

Theorem 8.5.3 The notion of equivalence of E -enriched categories is

closed under retracts and satisfies 3 for 2. If Af→ B and B

g→ A are

functors between E -enriched categories such that fg and gf are equiva-

lences, then f and g are equivalences.

Proof Suppose f : A → B is a retract of an equivalence of E -enriched

precategories, by a commutative diagram

A → U → A

B

f

↓

→ V

↓

→ B

f

↓

such that the horizontal compositions are the identity and the middle

vertical arrow is an equivalence. We get a corresponding diagram of sets

Isoτ≤1(A) → Isoτ≤1(U) → Isoτ≤1(A)

Isoτ≤1(B)

↓

→ Isoτ≤1(V )

↓

→ Isoτ≤1(B)

↓

where the middle vertical arrow is an isomorphism by Lemma 8.5.1. It

follows that Isoτ≤1(A) → Isoτ≤1(B) is an isomorphism, in particular f

is essentially surjective.

If x0, x1 ∈ Ob(A) is a pair of objects, denote the image objects in

U , V and B respectively by ui, vi and yi. Then we get a commutative

diagram

A(x0, x1) → U(u0, u1) → A(x0, x1)

B(y0, y1)

↓

→ V (v0, v1)

↓

→ B(y0, y1)

↓

in which the horizontal compositions are the identity and the middle ver-

tical map is an equivalence. The class of isomorphisms in E is closed un-

der retracts, by the previous lemma. It follows that A(x0, x1) → B(y0, y1)

is an isomorphism in E . This proves that f is fully faithful, completing


the proof that the class of equivalences between E -enriched categories is

closed under retracts.

Turn to the proof of the 3 for 2 property which says that if

Af→ B

g→ C

is a composable pair of morphisms and if any two of f , g and gf are

equivalences, then the third one is too.

Suppose that some two of f , g and gf are equivalences of E -enriched

categories. Applying the truncation functor gives a composable pair of

morphisms of sets

Isoτ≤1(A) → Isoτ≤1(B) → Isoτ≤1(C)

and the corresponding two of the maps are isomorphisms by Lemma

8.5.1. From 3 for 2 for isomorphisms of sets, it follows that the third

map is also an isomorphism, hence the third map among the f , g and

gf is essentially surjective.

For the fully faithful condition, consider first the two easy cases. If f

and g are global weak equivalences then for any pair of objects x0, x1 ∈

Ob(A) we have a factorization

A(x0, x1) → B(f(x0), f(x1)) → C(gf(x0), gf(x1))

where both maps are weak equivalences in E . The previous lemma gives

3 for 2 for isomorphisms in E , so the composed map is an isomorphism,

which shows that gf is fully faithful.

Similarly, if we assume known that g and gf are equivalences, then in

the same factorization we know that the composed map and the second

map are isomorphisms in E , so again by Lemma 8.5.2 it follows that the

first map is a weak equivalence, showing that f is fully faithful.

More work is needed for the third case: with the assumption that f and

gf are equivalences, to show that g is an equivalence. Applying Lemma

8.5.1 we get that Isoτ≤1(f) and Isoτ≤1(g) Isoτ≤1(f) are isomorphisms

of sets, so Isoτ≤1(g) is an isomorphism. The problem is to show the fully

faithful condition. Suppose x, y ∈ Ob(B). Choose x′, y′ ∈ Ob(A) and

isomorphisms in τ≤1(B), u ∈ τ≤1(B)(f(x′), x) and v ∈ τ≤1(B)(f(y′), y)

plus their inverses denoted u−1 and v−1. These are really maps u :

∗ → B(f(x′), x) and v : ∗ → B(f(y′), y) and similarly for u−1 and v−1.

The composition map

B(f(x′), x)× B(x, y)×B(y, f(y′)) → B(f(x′), f(y′))


composed with

u× 1× v−1 : ∗ ×B(x, y)× ∗ → B(f(x′), x) ×B(x, y)×B(y, f(y′))

gives

B(x, y) → B(f(x′), f(y′)).

Using u−1 and v gives B(f(x′), f(y′)) → B(x, y), and the associativity

axiom, definition of inverses and unit axioms combine to say that these

are inverse isomorphisms of objects in E . The action of g on morphism

objects respects all of these operations so we get a diagram

B(x, y) → C(g(x), g(y))

B(f(x′), f(y′))

↓

→ C(gf(x′), gf(y′)).

↓

The first vertical map is an isomorphism as described above. The second

vertical map is an isomorphism for the same reason applied to C and

noting that g(u) and g(v) are isomorphisms in τ≤1(C). The bottom map

is an isomorphism by Lemma 8.5.2 and because of the hypotheses that

f and gf are fully faithful. Therefore the top map is an isomorphism,

showing that g is fully faithful.

For the last part, suppose given functors between E -enriched cate-

gories Af→ B and B

g→ A such that fg and gf are equivalences. On

the level of sets τ≤0 we get a pair of maps whose compositions in both

directions are isomorphisms, it follows that τ≤0(f) and τ≤0(g) are iso-

morphisms. It also easily follows that for any objects x, y ∈ Ob(A), the

map

B(f(x), f(y)) → A(gf(x), gf(y))

has both a left and right inverse, so it is invertible. But since any object

of B is isomorphic to some f(x), an argument similar to the previous

one shows that B(u, v) → A(g(u), g(v)) is an isomorphism for any u, v ∈

Ob(B). Doing the same in the other direction we see that both f and g

are equivalences.


8.5.1 Interpretation of enriched categories as functors

∆oX → E

Lurie [153] has used an important variant on the notion of nerve of a

category (undoubtedly well-known in the 1-categorical context). This

makes the usual nerve construction apply to an enriched category, with-

out needing to assume anything further about E . In this point of view,

the set of objects is singled out as a set while the morphism objects

of an E -enriched category are considered as objects of E . The nerve is

then neither a functor from ∆o to Set, nor a functor to E , but rather

a mixture of the two. We will adopt this point of view when defining

M -enriched precategories later on. It has the advantage of allowing us

to avoid consideration of disjoint unions, sidestepping some of the diffi-

culties of [171].

If X is a set, define the category ∆X to consist of all sequences of

elements of X denoted (x0, . . . , xn) with n ≥ 0. One should think of such

a sequence as a decoration of the basic object [n] ∈ ∆. The morphisms of

∆X are defined in an obvious way generalizing the morphisms of ∆, by

just requiring compatibility of the decorations on the source and target.

This will be discussed further in Chapter 12.

If A is an E -enriched category, let X := Ob(A). The nerve of A is the

functor ∆oX → E , denoted also by A, defined by

A(x0, . . . , xn) := A(x0, x1)× · · · ×A(xn−1, xn).

Here the notations A(x, y) used on the right side are those of the enriched

category, but after having made the definition they are seen to be the

same as the notations for the nerve, so there is no contradiction in this

notational shortcut. The transition maps for the functor A : ∆oX → E

are obtained using the composition maps of A, for example

A(x0, x1, x2) = A(x0, x1)×A(x1, x2) → A(x0, x2)

in the main case that was discussed in the introduction.

Theorem 8.5.4 The category Cat(E ) of E -enriched categories be-

comes equivalent, via the above construction, to the category of pairs

(X,A) where X is a set and A : ∆oX → E is a functor satisfying the Se-

gal condition that for any sequence of elements x0, . . . , xn the morphism

A(x0, . . . , xn) → A(x0, x1)× · · · ×A(xn−1, xn)

is an isomorphism in E .


Understanding this theorem is crucial to understanding the weakly

enriched version which is the object of this book. It is left as an exercise

for the reader. Note that the definition of morphisms between pairs

(X,A) → (Y,B) is done in an obvious way, but the reader may consult

the corresponding discussion in Chapter 12 below.

8.6 Internal Hom

An important aspect of the theory is the cartesian condition on the

model categories involved. In this section, we explain how compatibility

with products induces an internal Hom.

Suppose M is a locally presentable category. Say that direct product

distributes over colimits if for any small diagram A : η →M and any

object B ∈M the natural map

colimi∈η(Ai × B) → (colimi∈ηAi)×B

is an isomorphism. If this is the case, then for any pair of diagrams

A : η → M and B : ζ →M , the natural map

colim(i,j)∈η×ζ(Ai ×Bj) → (colimi∈ηAi)× (colimj∈ζBj)

is an isomorphism, so these two ways of stating the condition are equiv-

alent.

We say that M admits an internal Hom if, for any A,B ∈ M the

functor E 7→ HomM (A×E,B) is representable by an object Hom(A,B)

contravariantly functorial in A and covariantly functorial in B together

with a natural transformation Hom(A,B)×A → B. That is to say that

a map E → Hom(A,B) is the same thing as a map A× E → B.

Proposition 8.6.1 Suppose M is a locally presentable category such

that direct product distributes over colimits. Then M admits an internal

Hom.

Proof See [1], Theorem 27.4, applying the fact that locally presentable

categories are co-wellpowered ([2, Remark 1.56(3)]).

Corollary 8.6.2 Suppose Φ is a small category. Then the category

of presheaves of sets Presh(Φ) = Func(Φp,Set) admits an internal

Hom. This may be calculated as follows: if A,B are presheaves of sets

on Φ then Hom(A,B) is the presheaf which to x ∈ Φ associates the set

HomPresh(Φ/x)(A|Φ/x, A|Φ/x).


Proof Note that Presh(Φ) is locally presentable. Direct products and

colimits are calculated objectwise, so direct product distributes over col-

imits since the same is true for the category Set. The explicit description

of Hom(A,B) is classical.

Corollary 8.6.3 Suppose M is a locally presentable category such

that direct product distributes over colimits, and suppose Φ is a small

category. Then direct product distributes over colimits in Func(Φ,M )

and this category admits an internal Hom.

Proof Again, direct product and colimits are calculated objectwise in

Func(Φ,M ).

8.7 Cell complexes

For the basic treatment we follow Hirschhorn [116]. Fix a locally pre-

sentable category M .

If α is an ordinal, denote by [α] the set α + 1, that is the set of all

ordinals j ≤ α. This notation extends the usual notation [n] used in

designating objects of the simplicial category ∆. Note that by definition

[α] is again an ordinal. We can write interchangeably i ≤ α or i ∈ [α].

A sequence is a pair (β,X·) where β is an ordinal, and X : [β] → M

is a functor. We usually denote this by the collection of objects Xi for

i ≤ β, with morphisms φij : Xi → Xj whenever i < j ≤ β. A sequence

is continuous if for any j ≤ β such that j is a limit ordinal, the map

colimj<jXi → Xj

is an isomorphism. A sequence yields in particular a morphismX0 → Xβ .

Suppose we are given a set of arrows I ⊂ Arr(M ). An I-cell complex

is a continuous sequence X : [β] →M together with the data, for each

i < β, of fi ∈ I and a diagram

Uifi→ Vi

Xi

ui

↓→ Xi+1

vi

↓

(8.7.1)

inducing an isomorphismXi∪UiVi ∼= Xi+1. In general, the choice of data

(fi, ui, vi)i≤β is not uniquely determined by the choice of sequence X·


although by abuse of notation we usually say just that Xii≤β is a cell

complex.

Denote by cell(I) the class of arrows in M which are of the form

X0 → Xβ for a cell complex indexed by [β].

8.7.1 Cell complexes in presheaf categories

We would like to develop the idea of a cell complex as corresponding to

a sequence of additions of cells, whereby one could notably envision to

change the order of attachment of the cells. It is useful to consider first

the special case when M is a presheaf category and the elements of I

are nontrivial injections of presheaves. In this case, an objects of M has

its own “underlying set” and the cells in a cell complex can be identified

with subsets. This case is much easier to understand, and it covers many

if not almost all of the examples we want to consider. We describe the

notion of inclusion of cell complexes in this special case. It is easier to

understand, but on the other hand this facility obscures what is required

for treating the general case, so this discussion should be considered as

optional.

Suppose Ψ is a small category, and M = Func(Ψ,Set). For A ∈M

define its underlying set to be

U(A) :=∐

x∈Ψ

A(x).

This construction gives a functor U : M → Set which is faithful and

compatible with colimits. A morphism in M is a monomorphism if and

only if it goes to an injection of underlying sets.

Suppose given a set of morphisms I in M , which we suppose to

be monomorphisms but not isomorphisms. Then, given a cell complex

Xii≤β , the set U(Xβ) − U(X0) is partitioned into nonempty subsets

which we call the cells, and which are indexed by the successor ordinals

i+1 ≤ β. The cell Ci+1 is by definition the complement U(Xi+1)−U(Xi).

Because of the assumption that elements of I are monomorphisms, all

of the transition maps in the cell complex are monomorphisms.

Associated to a cell Ci+1 is its attaching diagram (8.7.1) as above.

With those notations, in our case the map ui : Ui → Xi is uniquely

determined by the composed map Ui → Xβ.

If Yjj≤α is another cell complex, an inclusion of cell complexes con-

sists of morphisms Y0 → X0 and Yα → Xβ such that the morphism

U(Yα)− U(Y0) → U(Xβ)− U(X0)


is injective, respects the partitions, induces an isomorphism from each

cell for Y· to one of the cells for X·, and such that the attaching maps

for these corresponding cells are the same, where the attaching maps are

considered as maps into Yα and Xβ respectively.

A presheaf category will satisfy the hypotheses used by Hirschhorn

[116], so in this case we can refer there. The reader may note that start-

ing with a presheaf category M , our construction of the model category

PC(X,M ) keeps us within the realm of presheaf categories, and this

remark can be applied iteratively. So, in a certain sense for the main ex-

amples to be envisioned, the case of cell complexes in a presheaf category

is sufficient.

8.7.2 Inclusions of cell complexes

In spite of the previous remark, it seems like a good idea to search for

the highest convenient level of generality. Thus, we turn now to cell

complexes in the general situation where M is a locally presentable

category and I ⊂ Arr(M ) is a subset of morphisms.

In order to develop a theory allowing us to interchange the order of

cell attachment, we first define a notion of inclusion of cell complexes. An

intermediate approach was developped by Hirschhorn in [116] with his

notion of cellular model category. The general case has been treated by

Lurie in one of the first appendix to [153]. We give a somewhat different

discussion which is certainly less streamlined than Lurie’s, but we hope

it will help to gain an intuitive picture of what is going on.

Consider a strictly increasing map of ordinals q : [α] → [β]. Define

the map q− : [α] → [β] by

q−(i) := infk, ∀j < i, q(j) < k.

Thus q−(0) = 0 and for i > 0, q−(i) = supj<i(q(j) + 1). For successor

ordinals, q−(i + 1) = q(i) + 1, whereas if i is a limit ordinal then q−(i)

is the limit of q(j) for j < i. It follows that q− is strictly increasing, and

also continuous, which is to say that if i is a limit ordinal then q−(i) is

the limit of q−(j) for j < i. Furthermore q−(i) ≤ q(i). One can see from

the above characterization that the intervals

[q−(i), q(i)] := j ∈ [β], q−(i) ≤ j ≤ q(i)

are disjoint and cover [β], which is another way of understanding why

to introduce q−.

The basic idea of the following definition is that a cell complex is a


transfinite sequence of pushouts along specified morphisms in I, with

these specifications forming part of the data of the cell complex (in par-

ticular, a cell complex consists of more than just the resulting morphism

X0 → Xβ). An “inclusion of cell complexes” should mean a map com-

patible with the sequences of pushouts, via a map of ordinals which

should make the labels in I correspond.

Here is the precise definition. Suppose we are given two cell complexes

(α,X·) = (α,Xi, φij , fi, ui, vi) and (β, Y·) = (β, Yi, ψij , gi, ri, si). Here

gi : Ri → Si and ri : Ri → Yi with Siψi,i+1∪si

→ Yi+1.

An inclusion of cell complexes from (α,X·) to (β, Y·) is a pair con-

sisting of a strictly increasing map q : α → β, and a collection of mor-

phisms ξi : Xi → Yq−(i), subject to some conditions. Before explaining

the conditions, extend the map to q : [α] → [β] by setting q(α) := β

and notice that the maps ξi induce maps denoted ξi,+ : Xi → Yq(i),

defined by ξi,+ := ψq−(i)q(i) ξi because q−(i) ≤ q(i). This includes

ξα,+ : Xα → Yβ . In view of the relation q−(i + 1) = q(i) + 1, we have

ξi+1 : Xi+1 → Yq(i)+1. The conditions are as follows:

—first of all that gi = fq(i) as elements of the set I;

—second, if i is a limit ordinal then ξi is the colimit of the ξj for j < i,

going from Xi = colimj<iXj to Yq−(i) = colimj<iYq−(j);

—and third, for any i < α the composition

Riri→ Xi

ξi,+→ Yq(i)

Si

gi

↓si→ Xi+1

↓ξi+1→ Yq(i)+1

↓

is equal to the diagram

Uq(i)uq(i)→ Yq(i)

Vq(i)

fq(i)

↓ vq(i)→ Yq(i)+1

↓

for the complex Y·.

Suppose (γ, Z·) is a third cell complex and (p, ζ·) is a map of cell

complexes from (β, Y·) to (γ, Z·). We define the composition (p, ζ·)(q, ξ·)

to be the map of cell complexes (h, η·) given as follows. First of all,


h := p q : [α] → [γ]. Notice that h−(i) = p−(q−(i)), as can be seen

from the definitions of p− and q−. Thus it makes sense to define ηi to

be the composition

Xiξi→ Yq−(i)

ζq−(i)→ Zh−(i) = Zp−(q−(i)).

This collection of compositions satisfies the three conditions for being a

morphism of cell complexes.

Let Cell(M ; I) denote the category whose objects are I-cell com-

plexes in M and whose morphisms are the inclusions of cell complexes.

This is provided with functors s and t to M defined by s(α,X·) := X0

and t(α,X·) := Xα. If (q, ξ·) is an inclusion from (α,X·) to (β, Y·) then

ξα,+ : Xα → Yq(α) and we can compose with the transition map for Y·to get a map to Yβ . This defines the functoriality maps for t. The collec-

tion of maps X0 → Xα provides a natural transformation from s to t or

equivalently a functor a : Cell(M ; I) → Arr(M ) whose compositions

with the source and target functors are s and t respectively.

Recall that cell(I) denotes the class of arrows in the essential image

of a, in other words it is the class of morphisms f : X → Y in M such

that there exists (α,X·) ∈ Cell(M ; I) with X0 → Xα isomorphic to f

in Arr(M ).

Lemma 8.7.1 If (α,X·) is a cell complex and X0 → Y0 is a morphism,

then define Yi := Xi ∪X0 Y0. Using the induced maps for attaching data,

we get a cell complex (α, Y·). In particular, if

X → Y

Z

↓→ W

↓

is a cocartesian diagram in M such that the top arrow is in cell(I), then

the bottom arrow is also in cell(I). There is a tautological inclusion of

cell complexes (t, φ) : (α,X·) → (α, Y·) where t : [α] → [α] is the identity

and φi : Xi → Yi is the tautological inclusion.

Proof With the notations used in the definition of cell complex, given

that Xi ∪Ui Vi ∼= Xi+1 we get

Yi+1 = Y0∪X0Xi+1 = Y0∪

X0 (Xi∪UiVi) = (Y0∪

X0Xi)∪UiVi = Yi∪

Ui Vi.

Similarly Y· satisfies the continuity condition at limit ordinals.


Lemma 8.7.2 Given a continuous sequence indexed by an ordinal such

that the transition maps Xi → Xi+1 are in cell(I), the composition

X0 → colimiXi is again in cell(I).

Proof Just combine together the sequences.

Inclusions of cell complexes enjoy some rigidity properties because of

the fact that the map of labels is included as part of the data.

Lemma 8.7.3 Suppose q : [α] → [β] is a strictly increasing map, and

suppose that we are given two cell complexes (α,X·) and (β, Y·). For a

given initial map ξ0 : X0 → Y0, there is at most one inclusion of cell

complexes

(q, ξ·) : (α,X·) → (β, Y·),

based on q and starting with ξ0. If q and ξ0 are isomorphisms, and if

(q, ξ·) exists then it is an isomorphism.

Proof Suppose (q, ξ′·) is another inclusion of cell complexes with ξ′0 =

ξ0. We prove by induction that ξ′j = ξj for all j ≤ α. Suppose it is known

for all i < j. If j is a limit ordinal then the universal property of the

colimit Xj = colimi<jXi means that the map

ξj : Xj → Yq−(j)

is determined by the ξi for i < j, and the same is true of ξ′j . The inductive

hypothesis that ξ′i = ξi for i < j therefore implies ξ′j = ξj . This treats

the case of limit ordinals; the other possible case is assume j = i + 1.

Then the map ξj = ξi+1 fits into the diagram

Riri→ Xi

ξi,+→ Yq(i)

Si

gi

↓si→ Xi+1

↓ξi+1→ Yq(i)+1

↓

where the horizontal compositions are given as the attaching maps for

(β, Y·). On the other hand, Xi+1 is a pushout in the left square, that

is the left square is cocartesian. Thus ξi+1 is uniquely determined by

ξi,+ and the attaching map Si = Vq(i)vq(i)→ Yq(i)+1 which identifies with

ξi+1 si. The same determination holds for ξ′i+1, but ξi,+ is determined

by ξi so ξ′i,+ = ξi,+, therefore ξ

′i+1 = ξi+1. This completes the induction

step.


Suppose q and ξ0 are isomorphisms. Then α = β and q−(i) = q(i) = i

for all i ≤ α, and a similar induction shows that the ξi are all isomor-

phisms.

An application of the rigidity property will help with the uniqueness

part of the main accessibility result.

Corollary 8.7.4 Suppose (α,X·) is a cell complex, and suppose

(p, η·) : (β, Y·) → (γ, Z·)

is an inclusion of cell complexes. Suppose given a set A and a family of

inclusions of cell complexes

(q(a), ξ(a)) : (α,X·) → (β, Y·)

indexed by a ∈ A, such that the compositions (p, η·) (q(a), ξ(a)) are all

the same. Suppose furthermore that the maps X0ξ0(a)→ Y0 are all the

same. Then the (q(a), ξ(a)) are all the same.

Proof Since p is strictly increasing, it is injective. Thus, the condition

that the p q(a) are all the same implies that the q(a) are all the same;

the previous lemma immediately says that the (q(a), ξ(a)) are all the

same.

It is useful to have a different representation of an inclusion of cell

complexes (q, ξ·) : (α,X·) → (β, Y·). Define a family of objects denoted

Xj and indexed by j ∈ [β] as follows: let

Xj := Xi, q−(i) ≤ j ≤ q(i).

Note that for any j ∈ [β] there exists a unique i ∈ [α] such that q−(i) ≤

j ≤ q(i) (see one of the first properties of q− above). We have maps

Xj → Yj , and compatible transition maps Xj → Xk for any j ≤ k. For

exponents in the same sub-interval q−(i) ≤ j ≤ k ≤ q(i), the transition

maps are the identity.

In these terms, the cell attaching data consists of a subset c(X ·) ⊂ β,

defined as the set of all q(j) for j ∈ α, which we think of as the subset of

cells in X ·; plus, for each j ∈ c(X ·) two maps uj(X ·) and vj(X ·) fitting


into a diagram

Ujuj(X ·)

→ Xj → Yj

Vj

gj

↓vj(X ·)

→ Xj+1

↓

→ Yj+1

↓

where gj : Uj → Vj is the cell attached to Y· at j ∈ β and the horizontal

compositions are the attaching maps uj : Uj → Yj and vj : Vj → Yj for

Y·. For any j ∈ c(X ·) the above diagram has a cocartesian left square.

The composed outer square is cocartesian, being is the attaching diagram

for Yj → Yj+1, which implies that the right square is cocartesian also.

On the other hand, for j 6∈ c(X) the mapXj → Xj+1 is an isomorphism.

In the new notation note that the last element is Xβ = Xα. The maps

uj(X ·) and vj(X ·) are contained in the data of the cell complex (α,X·)

with appropriate renumbering via c(X ·) ∼= α.

The collection of data described above is the same as the inclusion of

cell complexes. Furthermore, compositions of inclusions of cell complexes

can be understood in this notation: if X · is a subcomplex of Y· then

the indexing ordinal α for X is isomorphic to the subset c(X ·) ⊂ β; a

subcomplex Z · of X · then consists of a subset

c(Z ·) ⊂ α ∼= c(X ·) ⊂ β,

and the composed inclusion of cell complexes from Z · to Y· corresponds

to the subset c(Z ·) ⊂ β obtained using transport of structure along the

isomorphism in the middle.

The category of ordinals with strictly increasing maps, isn’t closed

under sequential transfinite colimits. However, the category of ordinals

mapping to a given fixed ordinal, does admit sequential colimits. There-

fore the same is true of our notion of inclusion of cell complexes: whereas

Cell(M ; I) is not itself closed under sequential colimits, on the other

hand if we look at cell complexes included into a given (β, Z·), then we

can take sequential colimits.

Proposition 8.7.5 Suppose (β, Z·) ∈ Cell(M ; I) is a cell complex.

Then the category Cell(M ; I)/(β, Z·) is closed under filtered colimits:

if (α(k), X·(k)) is a family of cell complexes indexed by a filtered category

k ∈ ζ such that the transition maps are inclusions of cell complexes, all

provided with compatible inclusions (α(k), X·(k)) → (β, Z·), then there


is a colimit cell complex (α(ζ), Y·) again mapping to (β, Z·) and such

that Y0 = colimk∈ζX0(k) and Yα(ζ) = colimk∈ζXβ(k)(k).

Proof Use the alternative description of an inclusion of cell complexes

to (β, Z·). We obtain a family Xj(k) doubly indexed by j ∈ [β] and

k ∈ ζ. Put

Y j := colimk∈ζXj(k).

The maps Y j → Zj are given by the universal property of the colimit.

The subset of cells is defined as c(Y ·) :=⋃k∈ζ c(X

·(k)). If j ∈ c(Y ·),

choose kj ∈ ζ such that j ∈ c(X·(kj)). Then let kj\ζ be the category of

arrows kj → m in ζ. The functor kj\ζj → ζ is cofinal, so

Y j = colimm∈kj\ζXj(m).

For each kj → m we have the attaching maps

Uj → Xj(kj) → Xj(m), Vj → Xj+1(kj) → Xj+1(m),

which yield attaching maps

Uj → Y j , Vj → Y j+1.

If j 6∈ c(Y ·) then for any k ∈ ζ the map Xj(k) → Xj+1(k) is an

isomorphism, so the map Y j → Y j+1 is an isomorphism. If j is a limit

ordinal then

Xj(k) = colimi<jXi(k)

so passing to the double colimit, we have Y j = colimi<jYi. This gives

Y · the structure of cell complex with cell inclusion to Z · according to

our second description above.

This provides a colimit in the category Cell(M ; I)/(β, Z·), indeed if

(β′, Z ′·) is an object of the category provided with inclusions from all

the (X ·(k)), then we can understand the above construction as corre-

sponding to the same construction in Cell(M ; I)/(β′, Z ′·), which gives

the inclusion of cell complexes from Y · to (β′, Z ′·).

8.7.3 Cutoffs

Suppose (β, Z·) is a cell complex, and β′ ≤ β. Define the cutoff Cβ′(β, Z·)

to be the cell complex consisting of ordinal β′ and family of Zi for

i ≤ β′ ≤ β with the same structural data restricted to the subset of

values of i. If

(q, ξ·) : (α,X·) → (β, Z·)


is an inclusion of cell complexes, define its relative cutoff

Cβ′(q, ξ·) : Cα′(α,X·) → Cβ′(β, Z·)

as follows. Using the strictly increasing map q : α → β, put α′ :=

supi ≤ α, q(i) ≤ β′. Define a new map q′ equal to q on α, but extended

to [α] by setting q′(α) := β′. In general this is different from q(α).

Define the relative cutoff inclusion of cell complexes to be given by the

restricted map q′ : [α′] → [β′] and the family of maps ξi and associated

cell identification data for X· restricted to i ≤ α′.

In terms of the alternative notation for inclusions of cell complexes,

we denote the cutoff of a subcomplex X · by just Cβ′(X ·). In these terms,

the subset of cells is just the intersection

c(Cβ′(X ·)) = c(X ·) ∩ β′

and the attaching data are the same as those of X ·.

8.7.4 The filtered property for subcomplexes

In the context of our next main accessibility result for cell complexes,

we would like to consider the join of a set of inclusions of cell com-

plexes. Suppose (β, Z·) = (β, Z·, f·, u·, v·) is a cell complex, J is a set,

and (qj , ξ·(j)) : (αj , X·(j)) → (β, Z·) is a family of inclusions of cell

complexes. The “join” should be a cell complex sitting in the middle

(αj , X·(j)) → (ϕ, Y·) → (β, Z·).

The set of cells c(Y ·) should be the union of the sets of cells c(X·(j)) ⊂ β.

In order to start off the process, we need to make a choice of the place

to start Y0 which should fit into factorizations

Xj0 → Y0 → Z0

for all j. From there, one can proceed to attach the required cells, with-

out problem if M is a cellular model category [116]. This condition ba-

sically means that the arrows in I are monomorphisms enjoying good

properties; for example a presheaf category in which the cofibrations are

contained in the injections, is cellular.

In the general case, we run into the problem that the cell attaching

maps needed to construct Y · are not uniquely determined by those of

Z·. So, we proceed differently. The general situation has been treated by

Lurie in the appendix of [153]. Between the cellular case treated in [116]


and the general treatment in [153] our present discussion is undoubtedly

superfluous, included for completeness.

Suppose M is locally κ-presentable, that the elements of I are arrows

whose source and target are κ-presentable, and that I has < κ elements.

Let Cell(M ; I)κ denote the full subcategory of cell complexes (α,X·)

such that |α| < κ and X0 is κ-presentable. The following theorem lets

us replace the general notion of cofibration, by a cell complex. This was

done by Lurie in [153], A.1.5.12 so the reader could refer there instead.

Theorem 8.7.6 With the above hypotheses, suppose given a cell com-

plex (β, Z·) ∈ Cell(M ; I). Then the category Cell(M ; I)κ/(β, Z·) is

κ-filtered. Furthermore (β, Z·) is the colimit in Cell(M ; I) of the tau-

tological functor

Cell(M ; I)κ/(β, Z·) → Cell(M ; I)

and the arrow Z0 → Zβ is the colimit in Arr(M ) of the composition

of the tautological functor with Cell(M ; I) → Arr(M ).

Proof The objects of Cell(M ; I)κ/(β, Z·) are inclusions of cell com-

plexes (α,X·) → (β, Z·) such that |α| < κ and such that X0 is κ-

presentable.

We first show the uniqueness half of the κ-filtered property. Suppose

(p, η·) : (α,X·) → (β, Z·)

and

(p′, η′·) : (α′, X ′

· ) → (β, Z·)

are two objects of Cell(M ; I)κ/(β, Z·), and suppose given a set A of

cardinality |A| < κ indexing a family of morphisms from one to the

other in Cell(M ; I)κ/(β, Z·), that is to say a family of inclusions of

cell complexes

(q(a), ξ·(a)) : (α,X·) → (α′, X ′·)

such that (p′, η′·) (q(a), ξ·(a)) = (p, η·). Injectivity of p′ implies that the

q(a) are all the same. On the other hand we get the family of maps

ξ0(a) : X0 → X ′0

such that η′0 ξ0(a) = η0. Recall that X0 and X ′0 are required to be

κ-presentable. The same uniqueness part of the property that Mκ/Z0


is κ-filtered, says that the map X ′0 → Z0 factors as X ′

0

φ0→ Y0 → Z0

through a κ-presentable object Y0, such that all of the maps

φ0 ξ0(a) : X0 → Y0

are the same. As in Lemma 8.7.1, define a new cell complex (α′, Y·) by

setting

Yi := Y0 ∪X′

0 X ′i,

and keeping the same attaching data as for X ′· , and with the tautological

inclusion of cell complexes (t, φ·) from (α′, X ′·) to (α′, Y·). The condition

on the choice of Y0 together with the fact that the q(a) are all the same,

provide the hypotheses required in order to apply Corollary 8.7.4 to

conclude that all the maps of cell complexes

(t, φ·) (q(a), ξ·(a)) : (α,X·) → (α′, Y·)

are the same.

The elements Yi are κ-presentable, so (α′, Y·) ∈ Cell(M ; I)κ. The

map of cell complexes (p′, η′·) extends, using the pushout expressions for

Yi and the maps Y0 → Zi, to a map of cell complexes

(p′, ζ·) : (α′, Y·) → (β, Z·).

Via this map, we may consider (α′, Y·) as an element ofCell(M ; I)κ/(β, Z·).

We have

(p′, ζ·) (t, φ·) = (p′, η′·),

so (t, φ·) may be viewed as a map in Cell(M ; I)κ/(β, Z·). This gives

exactly a map there whose compositions with the (q(a), ξ·(a)) are all the

same, serving to prove the uniqueness half of the κ-filtered property.

For the remainder of the theorem, we prove the full statement by

induction on the length β of the cell complex Z. If β = 0 there is nothing

to prove. Hence, we may assume that the statement of the theorem is

known for all cell complexes (β′, Z ′·) with β

′ < β.

The main step is to prove that Cell(M ; I)κ/(β, Z·) is κ-filtered. Sup-

pose that X ·(a) is a collection of subcomplexes of (β, Z·) which are κ-

small, that is with X0 being κ-presentable and |c(X ·)| < κ, and indexed

by a set a ∈ A with |A| < κ. We have to show that they all map to a

single κ-small subcomplex Y ·.

For any β′ < β, consider the collection of cutoff complexes (using the

alternate notation at the end of Section 8.7.3) Cβ′(X ·(a)) mapping by

inclusions of cell complexes to Cβ′(β, Z·). By the induction hypothesis,


there is a single κ-small subcomplex Y ·(β′) of Cβ′(β, Z·), such that all

of the Cβ′(X ·(a)) map to Y ·(β′).

Assume that β is a limit ordinal approached by a sequence of β′ < β of

cardinality< κ. Then, by transfinite induction over this sequence we may

assume that the choice of Y ·(β′) is provided with transition inclusion

maps from Y ·(β′) to Y ·(β′′) whenever β′ ≤ β′′ are two members of the

sequence. Furthermore, we may take the colimit of all of the Y 0(β′) and

use this as starting element (it is a colimit of length < κ so it remains

κ-presentable). Thus we may assume that the Y 0(β′) are all the same, so

that Lemma 8.7.5 applies. Set Y · := colimβ′Y ·(β′). There are maps from

Cβ′(X ·(a)) to Y ·. The different maps from the X0(a) to Y 0, depending

on β′, compose to the same map into Z0. Since X0(a) is κ-presentable,

and the κ-presentable objects mapping to Z0 form a κ-filtered category

[2, Proposition 1.22], and also since both A and the sequence of β′ have

size < κ, we may choose W fitting into Y 0 → W 0 → Z0 but with W

still being κ-presentable. Then replace Y j by W ∪Y0

Y j . This way, all

of the maps X0(a) → Y 0 will be the same independent of β′. Then a

transfinite induction argument on the cutoff level β′ shows that the maps

Cβ′(X ·(a)) → Y · fit together in the colimit as β′ → β, to a collection

of maps X ·(a) → Y ·. This completes the proof of the filtered property

when β is a limit ordinal approached by a sequence of cardinality < κ.

If on the other hand β is a limit ordinal which is not approached

by any sequence of cardinality < κ, then there is some β′ < β such

that all of the cells in c(X ·(a)) are at j < β′ for all a ∈ A. Therefore

Cβ′(X ·(a)) = X ·(a), and these are cell complexes included in Cβ′(β, Z·).

The inductive hypothesis says that they all map to a same subcomplex

Y · of Cβ′(β, Z·), and this Y · serves to show the filtered property for the

family of cell complex inclusions X ·(a) → (β, Z·).

This leaves us with the case when β is a successor ordinal: β = η+1. It

is in some sense the main case where we need some work. This extra work

is occasionned by the goal of working without a monomorphism axiom

for the elements of I, such as was used by Hirschhorn [116] with his

notion of “cellular model category”. The basic problem is that when we

try to add in the last cell, the attaching maps may be ill-defined because

of various different collections of cell attachments up until then. Hence,

we need to backtrack and add some more cells so as to stabilise the

collection of attaching maps. For this we use the full inductive hypothesis

about the colimit over our filtered category, which applies to the cell

complex Cη(β, Z·) of length one less. This says that Zη is a κ-filtered

colimit of things obtained by inclusions of κ-small cell complexes.


To set things up more carefully, write A = A′∪A′′ where A′ consists of

those a such that X ·(a) involves the last cell, that is to say η ∈ c(X ·(a));

and A′′ consists of those a for which η 6∈ c(X ·(a)). One could suppose

that A′′ consists of a single element, indeed theX ·(a) which don’t involve

the last cell, are all subcomplexes of a single subcomplex of Cη(β, Z·)

by the inductive hypothesis, and we could take this one as the single

subcomplex indexed by A′′.

Now for each a ∈ A′ we have Xη(a) → Zη. Furthermore we have

attaching maps

uη(X·(a)) : Uη → Xη(a), vη(X

·(a)) : Vη → Xβ(a) ∼= Xη(a) ∪Uη Vη.

These lift the attaching maps uη : Uη → Zη and vη : Vη → Zβ. By the

inductive hypothesis, there is a single κ-small cell complex Y · mapping

by a cell complex inclusion to Cη(β, Z·), such that all of the Cη(X·(a))

for a ∈ A′, and all of the X ·(a) for a ∈ A′′, map to Y ·. Choose such

maps for each a, denoted ψ·(a). Then for any a ∈ A′ we get a composed

attaching map

uη(a) := ψη(a) uη(X·(a)) : Uη → Y η.

The composition of these maps with Y η → Zη are all the same, they

are the attaching map for the original complex Z·.

The inductive hypothesis tells us that Zη is a κ-filtered colimit of

W η as W · runs over the κ-filtered category of κ-small cell complexes

with inclusions to Cη(β, Z·). The category of objects W · under Y · is

cofinal in here, so we can view this colimit as being over factorizations

Y · → W · → Cη(β, Z·).

As, on the other hand, the category of κ-presentable objects mapping

to Zη is κ-filtered, there is a factorization Y η → T → Zη with T

being κ-presentable, such that all of the above maps uη(a) compose into

the same map Uη → T . Because Zη is a κ-filtered colimit of the W η,

the κ-presentable property of T tells us that there is a factorization

T → W η → Zη for some Y · → W · → Cη(β, Z·). Thus, all of the

composed maps

Uηuη(a)→ Y η → W η

are the same. We may now use this unique map as attaching map to

add on the last cell, to create a cell complex W · with an inclusion to

(β, Z·), whose cutoff at η is Cη(W·) = W ·. The identity of all of the

composed attaching maps provides us with inclusions of cell complexes

X ·(a) → W · for all a ∈ A (the cell attachment provides these inclusions


for a ∈ A′ and they are automatic for a ∈ A′′). This finishes the proof

of the κ-filtered property for Cell(M ; I)κ/(β, Z·).

To complete the proof of the inductive step we just have to note that

the colimit of the κ-small cell complex inclusions, which we now know is

κ-filtered and hence exists by Lemma 8.7.5, is equal to the full (β, Z·).

For this, note first of all that the colimit of all of the X0 will be Z0 by

the locally κ-presentable property of M . To conclude, it suffices to note

that every j ∈ β is a cell in at least one of the κ-small subcomplexes

X ·. If β is a limit ordinal, apply the inductive hypothesis saying that

we know the statement of the theorem for any β′ < β, and note that

for any j < β we can choose j < β′ < β. If β = η + 1 is a successor

ordinal, the inductive hypothesis gives the required statement for any

j < η so we may assume j = η. Then, argue as above. We know that

Cη(β, Z·) is a κ-filtered colimit of its κ-small subcomplexes, and as before

it follows that there will be a single κ-small subcomplex W · such that

the attaching map Uη → Zη factors through Uη → W η. The complex

W · obtained by attaching the last cell Uη → Vη onto W ·, satisfies the

current requirement that j = η be an element of c(W ·).

This completes the inductive proof of the theorem.

We now restate the result of the theorem in its simplified form which

will be used later for the pseudo-generating set construction of a model

category structure in Chapter 9.

Corollary 8.7.7 Suppose f : X → Y is a morphism in cell(I). Then

we can express f as a κ-filtered colimit of arrows fi : Xi → Yi, in

particular X = colimiXi and Y = colimiYi, such that Xi, Yi are κ-

presentable and fi ∈ cell(I) are cell complexes of length < κ.

Proof By definition of cell(I), there is a cell complex (β, Z·) ∈ Cell(M ; I)

such that f is equal to the map Z0 → Zβ . The theorem then says exactly

that f is a κ-filtered colimit of morphisms coming from cell complexes

in Cell(M ; I)κ.

We now describe how to fine-tune a map between colimits so that

it comes from a levelwise map of diagrams. Let M be a locally κ-

presentable category. Suppose α is a κ-filtered category, and F : α →M

is a diagram such that each Fi is κ-presentable. Suppose β is another

κ-filtered category and G : β →M is another diagram. Suppose given

a map

f : colimi∈αFi → colimj∈βGj .


Assume that the Fi are κ-presentable objects.

Lemma 8.7.8 In the above situation, there are a κ-filtered category

ψ and cofinal functors p : ψ → α and q : ψ → β and a natural col-

lection of maps fk : Fp(k) → Gq(k) depending on k ∈ ψ, i.e. a natural

transformation p∗F → q∗(G), such that the composition

colimi∈αFi = colimk∈ψFp(k)colimfk→ colimk∈ψGq(k) = colimj∈βGj

is equal to f .

Proof Let ψ be the category of triples (i, j, u) where i ∈ α, j ∈ β and

u : Fi → Gj is a morphism such that the diagram

Fi → colimi∈αFi

Gj

↓→ colimj∈βGj

↓

commutes. Since the Fi are κ-presentable, any i is part of a triple

(i, j, u) ∈ ψ. Any j sufficiently far out in β is also part of a triple, and

indeed ψ is κ-filtered, and the forgetful functors ψ → α and ψ → β are

cofinal. The third variable u provides the desired natural transformation

fk such that the diagram

colimk∈ψFp(k)∼=→ colimi∈αFi

colimk∈ψGq(k)

colimfk

↓ ∼=→ colimj∈βGj

f

↓

commutes.

The following proposition represents the idea that given an inclusion

of cell complexes, we can rearrange things so that the subcomplex comes

first and then the rest of the complex is added on later.

Proposition 8.7.9 If (α,X·) → (β, Y·) is an inclusion of cell com-

plexes, then there is another inclusion of cell complexes (δ, Z·) → (β, Y·)

with Z0 = Xα ∪X0 Y0 and Zδ = Yβ, such that β is the disjoint union of

c(X ·) and c(Y ·).

Proof Left to the reader.

8.8 The small object argument 155

8.8 The small object argument

Suppose A ⊂ Arr(M ) is a class of morphisms. We say that a morphism

f : X → Y satisfies the right lifting property with respect to A if, for

any commutative diagram

Ua→ X

V

u

↓b→ Y

f

↓

such that u ∈ A, there exists a lifting s : V → X such that fs = b and

su = a. We say that f satisfies the left lifting property with respect to A

if, for any commutative diagram

Xa→ U

Y

f

↓b→ V

v

↓

such that v ∈ A, there exists a lifting s : Y → U such that vs = b and

sf = a.

If I ⊂ Arr(M ) is a subset of morphisms, we have defined above

cell(I) ⊂ Arr(M ) to be the class of morphisms f : X → Y such that

there exists an I-cell complex (β,X·) with X0 = X and Xβ = Y , with f

being the transition map from X0 to Xβ. The class cell(I) defined this

way is clearly closed under compositions.

Let inj(I) ⊂ Arr(M ) be the class of maps which satisfy the right

lifting property with respect to cell(I), and let cof(I) ⊂ Arr(M ) be

the class of maps which satisfy the left lifting property with respect to

inj(I). Note that cell(I) ⊂ cof(I). The famous small object argument

as it applies to locally presentable categories, can be summed up in the

following theorem.

Theorem 8.8.1 Suppose M is a locally presentable category, and I ⊂

Arr(M ) is a small subset of morphisms. Then:

—any morphism f : X → Y admits a factorization f = pg where

Xg→ Z

p→ Y , such that g ∈ cell(I) and p ∈ inj(I);

—one may choose a factorization which is functorial in f ;


—the class cof(I) is closed under retracts and is equal to the class of

morphisms f : X → Y such that Y is a retract of some g : X → Z in

cell(I), in the category X\M of objects under M ;

—the class inj(I) is also equal to the class of morphisms which satisfy

the right lifting property with respect to cof(I).

Proof Refer to the numerous discussions in the literature.

We record here some basic facts about lifting properties.

Lemma 8.8.2 Suppose M is a category and A is a class of arrows in

M . Then the class F of morphisms in M which satisfy the right lifting

property with respect to all morphisms of A , is closed under retracts.

Similarly, if B is a class of arrows then the class G of morphisms which

satisfy the left lifting property with respect to all morphisms of B, is

closed under retracts.

Proof Suppose f : X → Y is in F , and g : A → B is a retract of f

with a retract diagram

Ai→ X

r→ A

B

↓j→ Y

↓s→ B.

↓

Suppose

Ua→ A

V

u

↓b→ B

g

↓

is a diagram with u ∈ A . Compose with the left square of the retract

diagram to get

Uia→ X

V

u

↓jb→ Y.

f

↓

The lifting property for f says that there exists t : V → X with tu = ia


and ft = jb. Composing with r gives a lifting rt : V → A such that

rtu = ria = a and grt = sft = sjb = b, so rt is a lifting for the original

diagram to g.

The proof for G is similar.

Corollary 8.8.3 Suppose I is a set of morphisms in a locally pre-

sentable category M . Then the class inj(I) is closed under retracts.

Proof The class inj(I) is defined by the right lifting property with

respect to cell(I).

Lemma 8.8.4 Suppose M is a complete and cocomplete category, and

A is a class of arrows in M . Then the class F of morphisms in M

which satisfy the right lifting property with respect to all morphisms of

A , is closed under fiber products and sequential inverse limits. Similarly,

if B is a class of arrows then the class G of morphisms which satisfy the

left lifting property with respect to all morphisms of B, is closed under

pushouts and transfinite composition.


8.9 Injective cofibrations in diagram categories

We turn now to two of the main results from the appendix to Lurie’s

[153]; these are notably refered to by Barwick [16] who gives a gen-

eral discussion of the injective model structure on section categories of

left Quillen presheaves. We need these results in order to construct in-

jective model categories in what follows, and we need to understand

something about the proof in order to apply it to the case of unital

diagram categories where we require that some values are equal to ∗.

The novice reader is invited to skip this section, refering to [153] for the

results, and imagining their extension to the unital case. Alternatively,

the arguments can be made more concrete in the case (due originally to

Heller [113]) where M is a presheaf category and the cofibrations are

monomorphisms.

Fix a locally κ-presentable category M and a subset I of morphisms.

Let cof (I) denote the class of I-cofibrations, that is morphisms which

are retracts of morphisms in cell(I). Let cof (I)κ denote a set of repre-

sentatives for the isomorphism classes of morphisms A → B in cof(I)

such that A and B are κ-presentable. Lurie’s first theorem [153, A.1.5.12]

is:


Theorem 8.9.1 In the above situation, cell(cof (I)κ) = cof(cof (I)κ) =

cof(I), that is to say any I-cofibration can be expressed as a cell complex

whose attaching maps are taken from the set cof(I)κ.

Proof The inclusions cell(cof (I)κ) ⊂ cof (cof (I)κ) ⊂ cof (I) are im-

mediate, we need to show that cof (I) ⊂ cell(cof (I)κ). Suppose w :

A → V is in cof (I). Using the small object argument, we can choose a

diagram

B

A →

f

→

V

p

↓

s

↑

with a map Af→ B in cell(I), a projection B

p→ V in inj(I), and a

section Vs→ B compatible with the map from A, such that ps = 1V .

Let πB := sp : B → B be the idempotent π2B = πB, compatible with

the identity map of A.

For the transfinite induction step, it will be convenient to consider

more generally the case when πB is compatible with πA which can be a

nontrivial idempotent on A.

The idea is to find κ-presentable cell complexes fi : Ai → Bi included

into f , together with definitions of πB,i and πA,i on the source and

target. Note that, given an idempotent πB,i : Bi → Bi we can let Vi be

the colimit of the diagram

BiπB,i→ Bi

πB,i→ Bi

πB,i→ . . . ,

then we have a projection pi : Bi → Vi and the system of maps πi on the

colimit gives si : Vi → Bi with pisi = 1Ui . Similarly let Ui be the image

of πA,i defined in the same way, so if Ai → Bi are in cell(I) then their

retracts Ui → Vi are in cof(I) since cof(I) is closed under retracts.

In order to construct the fi, proceed as follows. Let

ξ := Cell(M ; I)κ/f

denote the category of inclusions of cell complexes over f . By Theorem

8.7.6, ξ is κ-filtered. For g ∈ ξ let h(g) : X(g) → Y (g) denote the corre-

sponding cell complex, mapping to f by an inclusion of cell complexes.

We have A = colimg∈ξX(g), B = colimg∈ξY (g), and f is the colimit of

the g ∈ ξ. Let x(g) : X(g) → A and y(g) : Y (g) → B denote the maps


to the colimits. On the other hand, X(g) and Y (g) are κ-presentable. In

particular, the map

πBy(g) : Y (g) → B

has to factor through a map πB(g) : Y (g) → Y (n(g)) for some g → n(g) ∈

ξ which is a function of g. Furthermore for any v : g → h in ξ there

is n(g), n(h) → n′(v) such that the projections of πB(g) and π(h) into

Y (n′(v)) are the compatible along Y (v), and finally there is n(n(g)) → n′′(g)

such that πB(n(g))π(g) = πB(g) after projecting into Y (n′′(g)). Choose

similarly πA(g) and assume that the same choices work for πA(g), and

furthermore that πB(g)h(g) = πA(g) after projection to Y (n(g)).

A good filtered subcategory ξi ⊂ ξ is a full subcategory with < κ ob-

jects, filtered (note however that it would be too small to be κ-filtered),

and such that for any g ∈ ξi the elements n(g), n′′(g) and their arrows

are also in ξi; and for an arrow v in ξi the object n′(v) with its arrows

is also in ξi. If ξi is a good filtered subcategory, then

Bi := colimg∈ξiY (g)

has an endomorphism πB,i : Bi → Bi defined by using the πB(g).

Similarly πA,i : Ai → Ai is defined using the πA(g); these are compatible

and π2i = πi. By Proposition 8.7.5, fi is in cell(I), so the images Ui

(resp. Vi) of the idempotents πB,i (resp. πA,i) as defined above, have an

I-cofibration wi : Ui → Vi. Note that wi ∈ cof (I)κ.

Recall that by a cell of f we mean an element of the ordinal indexing

the cell attachements. An inclusion of cell complexes generates a corre-

sponding subset of cells, although in the general case we are currently

considering, specification of the subset of cells is not sufficient to spec-

ify the inclusion of cell complexes, because our attaching maps are not

necessarily monomorphisms.

For any given cell of f , we can choose a good filtered subcategory such

that the inclusion of cell complexes from Bi to B, contains the given

cell. Hence A ∪Ai Bi is a pushout of A along an element wi ∈ cof(I)κcontaining the given cell.

Start now with our given situation w : A → V being a retract of f :

A → B. We construct by transfinite induction, a sequence of inclusions of

cell complexes A → Cj → B together with definitions of the idempotent

πC,j on Cj , compatible with 1A and πB . Start with C0 = 1. Assume Cjis chosen for j < j0. If j0 is a limit ordinal then let Cj := colimj<j0Cj .

If not, j0 = k + 1 and we are in the general situation envisioned above:

Ck has its idempotent πC,k compatible with πB via the cell complex bk :


Ck → B. Fix the smallest cell of B not contained in the subset of cells of

Ck, and choose according to the previous procedure applied to the map

bk, a κ-presentable cell complex wj : Aj → Bj with an inclusion of cell

complexes from wi to (bk : Ck → B), and with compatible idempotents

πA,j and πB,j . Set

Cj = Ck+1 := Ck ∪Aj Bj ,

with its induced idempotent which will be called πC,j .

The process stops when there are no longer any cells of B but not in

Ck, that is to say Ck = B. Now let W0 := A and let Wj be the image of

the idempotent πC,j defined as

Wj := colim(Cj

πC,j→ Cj

πC,j→ . . .

).

If Uj and Vj denote the images of πA,j and πB,j respectively, we have

Uj → Vj ∈ cof (I)κ. Note that

Wj =Wk ∪Uj Vj

by commutation of colimits. Similarly if j is a limit ordinal then Wj =

colimi<jWi. The sequence Wj is an expresssion of f : A → V as an

element of cell(cof (I)κ).

Lurie’s second theorem is the application to diagram categories. Keep-

ing the previous notations, suppose Φ is a κ-small category (i.e. its ob-

ject and morphism sets have cardinality < κ). Within the category

Func(Φ,M ) say that a map A → B is an injective cofibration if

A(x) → B(x) is in cof(I) for any x ∈ Ob(Φ). This class of morphisms

is denoted Func(Φ, cof(I)). Let Func(Φ, cof (I))κ denote a set of rep-

resentatives for isomorphism classes of injective cofibrations between κ-

presentable objects, which is also the set of morphisms A → B such

that each A(x) → B(x) is in cof(I)κ. The following statement is [153,

A.2.8.3]; the argument is similar to the previous one.

Theorem 8.9.2 In the above situation,

Func(Φ, cof (I)) = cof (Func(Φ, cof (I))κ),

i.e. the set Func(Φ, cof (I))κ generates the class of injective cofibrations.

Proof In light of the previous theorem, we may assume that cof(I) =

cell(I). Suppose f : A → B is in Func(Φ, cof (I)), then each f(x)

can be given a structure of I-cell complex. Make such a choice for each


x ∈ Ob(Φ), although these choices are not compatible with the structure

of diagram.

Let ξ(x) be the κ-filtered category of κ-presentable inclusions of cell

complexes into f(x) (see Theorem 8.7.6). For i ∈ ξ(x) we have a κ-

presentable cell complex f(x, i) : A(x, i) → B(x, i) with an inclusion to

f , and B(x) = colimi∈ξ(x)B(x, i). If i ∈ ξ(x) and ϕ : x → y is an arrow

in Φ then there is j ∈ ξ(y) plus a lifting to a map B(x, i) → B(y, j)

compatible with B(ϕ). We can assume that the same j works for any ϕ;

for x, y, z and i ∈ ξ(x) we can choose k ∈ ξ(z) such that the compositions

of any transition maps from B(x, i) to B(y, j) then to B(z, k) satisfy the

product rule. Continuing in this way, we obtain a collection of operations

defined on⋃x ξ(x) together with lifting data, such that if ξ′(x) ⊂ ξ(x) is

a collection of filtered subcategories preserved by these operations, then

the collection of B′(x) := colimi∈ξ′(x)Bi is given a structure of diagram

B′. Similarly for A′ and the map A′ → B′ is an injective cofibration.

We can choose ξ′(x) with cardinality < κ, so A′ → B′ is a κ-presentable

injective cofibration, and in such a way that B′(x) contains any given

cell of some B(x0).

Now applying the same inductive argument as at the end of the pre-

vious theorem, which we don’t repeat, gives an expression of A → B as

a transfinite composition of pushouts along such A′ → B′.

The version we really need later is a variant in which the diagram cate-

gory is replaced by the unital diagram category denoted Func(Φ/Φ0,M ),

discussed in more detail in Chapter 13. Here Φ0 is a subset of objects of

Φ and A ∈ Func(Φ/Φ0,M ) means that A : Φ →M is a diagram such

that A(x) ∼= ∗ is the coinitial object of M , for all x ∈ Φ0. As before if I

is a fixed set of maps in M , a map A → B in this category is an injective

cofibration if each A(x) → B(x) is in cof (I). Note that isomorphisms

are always contained in cof(I) so this condition holds automatically

whenever x ∈ Φ0.

Theorem 8.9.3 The class Func(Φ/Φ0, cof(I)) of injective cofibra-

tions in the unital diagram category, equals cof (Func(Φ/Φ0, cof(I))κ),

i.e. it is generated by the set Func(Φ, cof(I))κ of representatives for

isomorphism classes of κ-presentable injective cofibrations.

Proof Follow the same procedure as in the previous theorem, noting

that all coproducts involved are taken over connected categories, and

by Lemma 8.1.8 the unitality condition is preserved by such coproducts.

The κ-presentable objects of Func(Φ/Φ0,M ) are the unital diagrams


A which are κ-presentable as diagrams, or equivalently such that each

A(x) is κ-presentable in M . So, the process described in the proof of

the previous theorem leads to an expression of an arbitrary injective

cofibration as a transfinite composition of pushouts along κ-presentable

ones, and all of the objects intervening here remain unital.

9

Model categories

In this chapter, we recall some of the basic elements of Quillen’s theory of

model categories and modern variants. We consider model structures on

diagram categories, Quillen functors, and left Bousfield localization. One

of the goals is to understand a quick version of left Bousfield localization

which is easier to understand than the general version, but which needs

a restrictive collection of hypotheses. It turns out that these hypotheses

will hold in the cases we need.

Along the way, we also discuss some abstract notions in category the-

ory which are useful for formulating the small object argument.

As pointed out in the previous chapter, the motivation for introducing

Quillen’s theory into the world of n-categories is the fact that there is a

localization problem at the heart of the route towards construction of the

n+1-category nCAT of all n-categories. Grothendieck was undoubtedly

aware of this connection at least on an intuitive level, because “Pursuing

Stacks” started out as a series of letters to Quillen.

9.1 Quillen model categories

Start by recalling Quillen’s definition of a closed model category. This

is a category M provided with three classes of morphisms, the “weak

equivalences”, the “cofibrations”, and the “fibrations”. The intersection

of the classes of cofibrations and weak equivalences is called the class

of “trivial cofibrations”, and similarly the intersection of the classes of

fibrations and weak equivalences is called the class of “trivial fibrations”.

Quillen asks that these should satisfy the following axioms.

(CM1)—The category M should be closed under finite limits and colim-

its. Following modern tradition, we really require that it be closed under


164 Model categories

all small limits and colimits.

(CM2)—The class of weak equivalences satisfies 3 for 2: given a com-

posable sequence of arrows in M

Xf→ Y

g→ Z

if any two of f , g and gf are weak equivalences, then so is the third one.

(CM3)—The classes of cofibrations, fibrations, and weak equivalences

should be closed under retracts: given a diagram

Ai→ X

r→ A

B

↓j→ Y

↓s→ B

↓

such that ri = 1A and sj = 1B, such that the two outer downward ar-

rows are the same, if the middle arrow g : X → Y is a cofibration (resp.

fibration, weak equivalence) then the outer arrow f : A → B is also a

cofibration (resp. fibration, weak equivalence).

(CM4)—Cofibrations satisfy the left lifting property with respect to triv-

ial fibrations, and trivial cofibrations satisfy the left lifting property with

respect to fibrations.

(CM5)—If f : X → Y is any morphism, then there exist factorizations

(i) and (ii) of f as the composition Xg→ Z

p→ Y such that

(i) g is a cofibration and p is a trivial fibration;

(ii) g is a trivial cofibration and p is a fibration.

It follows from these axioms that any two of the classes of cofibrations,

fibrations and weak equivalences, determine the third. For example a

morphism is a fibration (resp. trivial fibration) if and only if it satisfies

right lifting with respect to any trivial cofibration (resp. cofibration),

and dually. A morphism is a weak equivalence if and only if it factors

as a composition of a trivial fibration and a trivial cofibration, these

classes being determined from the classes of cofibrations and fibrations

respectively by the lifting property. Each of the three classes contains

any isomorphism.

We point out, as was done in [193], that the diagram included in [175]

for the definition of “retract” is visibly wrong, so his notion of “retract”

is not well defined. There could be two reasonable interpretations of this

condition. For condition (CM2) we have adopted the weak interpreta-

tion. The stronger interpretation would have the arrows on the bottom

9.1 Quillen model categories 165

row going in the opposite direction. If f is a strong retract of g then it is

also a weak retract of g. Hence, closure under retracts as we require in

(CM2) also implies closure under strong retracts. This choice coincides

with what was said in Dwyer-Spalinski [93, 2.6]. Similarly, Hinich [115]

uses the retract condition stated as we have done above.

Lemma 9.1.1 If M is a closed model category i.e. satisfies (CM1)–

(CM5), then the classes of trivial cofibrations and fibrations determine

each other by the lifting property; and similarly the classes of cofibrations

and trivial fibrations determine each other. In other words, a morphism

is a fibration (resp. trivial fibration) if and only if it satisfies the right

lifting property with respect to all trivial cofibrations (resp. all cofibra-

tions). And a morphism is a cofibration (resp. trivial cofibration) if and

only if it satisfies the left lifting property with respect to all trivial fibra-

tions (resp. all fibrations).

Proof There are four things to prove. Consider for example the fact

that a morphism is a fibration if and only if it satisfies the right lifting

property with respect to trivial cofibrations; the other three arguments

are identical. If f is a fibration then by (CM4) it satisfies the right lifting

property with respect to all trivial cofibrations. Suppose on the other

hand that f is a morphism which satisfies the right lifting property with

respect to all trivial cofibrations. Using (CM5), factor f = pg,

Xg→ Z

p→ Y

where g is a trivial cofibration and p is a fibration. Apply the right lifting

property being assumed for f , to the diagram

X=→ X

Z

g

↓p→ Y

f

↓

to get a morphism s : Z → X such that sg = 1X and fs = p. Putting s


into the diagram

Xg→ Z

s→ X

Y

↓=→ Y

↓=→ Y

↓

now says that f is a retract of the fibration p : Z → Y , so by (CM3) f

is a fibration.

9.2 Cofibrantly generated model categories

There is an important class of model categories which are particularly

easy to work with, and which contains many if not most of the examples

currently considered as important. Useful examples of model categories

which are not cofibrantly generated, will often be Quillen equivalent

to cofibrantly generated ones. On the other hand, this notion is very

helpful for a number of the operations we need, such as taking the model

category of M -diagrams, and left Bousfield localization. The reader is

referred to Hirschhorn’s excellent book [116] for a full explanation of

everything concerning cofibrantly generated model categories.

Suppose J ⊂Mor(M ) is a small subset of morphisms, then we define

classes of morphisms cell(J), inj(J) and cof (J), see Sections 8.7.2 and

8.8.

Recall that cof (J) consists of arrows X → Y which are retracts, in

the category of objects under X , of elements of cell(J). This condition

means that there should exist X → Z in cell(J) and maps Y → Z → Y

compatible with the maps from X and composing to the identity of Y .

Recall that a Umodel category M is cofibrantly generated if it is closed

under U-small limits and colimits, and if there are U-small subsets of

arrows I, J ⊂ Arr(M ) satisfying the following properties:

(CG1) the sources and targets of arrows in I and J are small in M ;

(CG2a) the arrows in I are cofibrations;

(CG2b) the trivial fibrations of M are the morphisms satisfying lifting

with respect to arrows in I;

(CG3a) the arrows in J are trivial cofibrations;

(CG3b) the fibrations of M are the morphisms satisfying lifting with

respect to arrows in J .

9.3 Combinatorial and tractable model categories 167

As a matter of notation, we say that (M , I, J) is a cofibrantly gener-

ated model category, if M is a model category and I and J are subsets

of arrows satisfying the above axioms.

This notation is convenient in that I and J determine the model cat-

egory structure, indeed I determines the class of trivial fibrations by

(CG2b) and J determines the class of fibrations by (CG3b). These in

turn determine respectively the classes of cofibrations and trivial cofibra-

tions by the saturated lifting properties. Then the class of weak equiva-

lences is determined by either one of the factorization properties.

Given a triple (M , I, J) consisting of a category admitting U-small

limits and colimits, and two subsets of arrows, we can try to define a

model structure following the recipe of the previous paragraph. If these

classes of (trivial) cofibrations, (trivial) fibrations and weak equivalences

do in fact form a closed model structure, and if (CG1) is satisfied, then

(M , I, J) is a cofibrantly generated model category.

See [116] [103] and other references, for discussions of various recogni-

tion properties telling when a triple (M , I, J) yields a cofibrantly gener-

ated model category. For example, the following statement will be useful.

Proposition 9.2.1 If M is a locally presentable category and we start

with the three classes of cofibrations, fibrations and weak equivalences;

then define trivial cofibrations as the intersection of cofibrations and

weak equivalences and similarly for trivial fibrations; if I and J satisfy

properties (CG1)–(CG3b); and if furthermore we know that weak equiva-

lences are closed under retracts and satisfy 3 for 2, then it is a cofibrantly

generated model category.

Proof Indeed, (CM5) comes from the small object argument, (CM4)

comes from the hypotheses (CG2b) and (CG3b), (CM3) is supposed for

weak equivalences and follows for inj(J) and cof (I), and (CM1) and

(CM2) are supposed.

9.3 Combinatorial and tractable model categories

It is most convenient to combine the notion of cofibrantly generated

model category with a good category-theoretical condition on M which

guarantees the small object argument. As was observed by J. Smith and

reported by D. Dugger in several papers for example [84], the appropriate

condition is to require that M be locally presentable.


A combinatorial model category is a cofibrantly generated model cat-

egory which is also locally presentable. In this case all objects are small

in M so (CG1) is automatic.

Barwick then refined this by requiring that the domains of generating

cofibrations and trivial cofibrations also be cofibrant: a combinatorial

model category is tractable if in addition for the given generating sets I

and J :

(TR)—the sources of arrows in I and J are cofibrant.

As we shall discuss below, diagrams with values in a combinatorial

model category have injective and projective model structures. If the

target category is tractable, then the projective model structure on di-

agrams is also tractable; it isn’t clear whether this is known in the case

of the injective structure.

9.4 Homotopy liftings and extensions

For X ∈M a cylinder object is a diagram

X ∪∅ Xi0∪i1→ C

p→ X

such that i0∪ i1 is a cofibration and p is a trivial fibration. The existence

is guaranteed by Axiom (CM5)(i). Similarly, if X → Y is a cofibration

there exists a relative cylinder object which is a diagram

Y ∪X Yi0∪i1→ C

p→ Y

again with i0 ∪ i1 a cofibration and p a trivial fibration. Quillen shows

that if X is cofibrant and A fibrant, then two maps f0, f1 : X → A

are homotopic, that is project to the same map in ho(M ), if and only

if there is a cylinder object and a map (or for any cylinder object there

is a map) C → A inducing the two given maps. Given a cofibration

X → Y and two maps f0, f1 : Y → A which agree when restricted to

X , we say they are homotopic relative to X if there is a relative cylinder

object and a map C → A inducing the two maps.

Weak equivalences between fibrant objects can be characterized by a

homotopy lifting property, being careful to look at homotopies relative to

the subobject of the cofibration. This is a classical fact from the theory

of model categories but, as it is a technical step needed in Chapter 13,

the proof is included here for completeness.

9.4 Homotopy liftings and extensions 169

Lemma 9.4.1 Suppose g : X → Y is a map between fibrant objects in

a tractable model category M , such that for any diagram

Uu→ X

V

f

↓v→ Y

g

↓

where f is a generating cofibration, there exists a lifting r : V → X such

that rf = u, together with a relative cylinder object

V ∪U Vj0∪j1→ IV

q→ V

with j0 ∪ j1 a cofibration and q a weak equivalence, and a map h :

IV → Y restricting to gu = vf on U , such that hj0 = gr and hj1 = v.

Then g is a weak equivalence.

Proof Factor Xk→ X ′ g′

→ Y where k is a trivial cofibration and g′ is

a fibration. Since X is assumed fibrant there is a retraction s from X ′

to X with sk = 1X . We show that g′ is a trivial fibration by the lifting

property. Suppose given a diagram

Uu′→ X ′

V

f

↓v′→ Y

g′

↓

with f a generating cofibration. Put u := su′ : U → X ; there is a

homotopy between ku and u′, given by a cylinder object

U ∪∅ Ui0∪i1→ IU

p→ U

with a map z : IU → X ′ with zi0 = ku and zi1 = u′. Choose a

compatible cylinder object for V , fitting into a diagram

U ∪∅ Ui0 ∪ i1

→ IUp→ U

V ∪∅ V

f

↓l0 ∪ l1

→ I ′V

If

↓q→ V.

f

↓


Extend the map IU∪i1,U,fV → Y given by g′z∪v′, to a map t : I ′V → Y

(this is an extension along a trivial cofibration, for maps to the fibrant

object Y ). Restricting along l0 now gives a map v : V → Y such that

vf = g′ku = gu. We obtain a diagram

Uu→ X

V

f

↓v→ Y.

g

↓

By hypothesis there is a lifting r : V → X such that rf = u and gr

is homotopic to v relative to U . This gives a lifting kr : V → X ′ such

that krf = ku and g′kr is homotopic to v relative to U . Putting this

back together with the previous homotopy, we get the top map in the

diagram

IU ∪i0,U,f Vz ∪ kr

→ X ′

I ′V

If ∪ l0

↓t→ Y

g′

↓

where the left map is a trivial cofibration and g′ is a fibration. This di-

agram doesn’t commute, however by the hypothesis of the lemma (and

using the notations from there) there is a homotopy relative to IU mak-

ing it commute. Adding this on and shifting the map from the component

V on the upper left, to the other side of the new cylinder object IV ,

gives the commutative diagram

IU ∪i0,U,f Vz ∪ kr

→ X ′

I ′V ∪l0,V,j0 IV

If ∪ j1

↓t ∪ h

→ Y

g′

↓

where the left map remains a trivial cofibration. Hence there exists a

lifting I ′V ∪l0,V,j0 IV → X ′ which, when restricted along l1 gives the

desired lifting to show that g′ is a trivial fibration. This completes the

proof.

A similar consideration holds for extensions.


Lemma 9.4.2 Suppose given two cofibrations Xa→ Y

b→ Z and maps

Yf→ A and Z

g→ A to a fibrant object A, such that fa = gba. Suppose

that gb is homotopic to f relative to X. Then there exists a map Zg′

→ A

such that g′b = f .

Proof Let C be a relative cylinder object for X → Y . Choose a factor-

ization

Z ∪Y,i0 C ∪Y,i1 Zd→ D

q→ Z

of a trivial fibration composed with a cofibration. Then D is also a rela-

tive cylinder object for X → Z and we have the compatibility diagram

Y ∪X Yi0 ∪ i1

→ Cp→ Y

Z ∪X Z

b ∪ b

↓j0 ∪ j1

→ D

c

↓q→ Z.

b

↓

Choose a map Ch→ A such that hi0 = f and hi1 = gb. We get a map

C ∪Y,i1 Zh∪g→ A

but the cofibration Yi1→ C is a weak equivalence, so the pushout along

i1 is a trivial cofibration Z → C∪Y,i1Z. Using the fact that d was chosen

to be a cofibration at the start, it follows that

C ∪Y,i1 Zd′

→ D

is a trivial cofibration. The fibrant condition for A now allows us to

extend the previous map h ∪ g along d′, so we get a map Dh′

→ A

extending h. The restriction g′ := h′j0 : Z → A provides the required

map.

9.5 Left properness

Recall that a model category M is left proper if, in any pushout square

X → Y

Z

↓→ W

↓


such that X → Y is a cofibration and X → Z is a weak equivalence,

then Y → W is also a weak equivalence.

Lemma 9.5.1 Suppose M is a left proper model category, and suppose

we are given a diagram

X → Y

Z

↓V

↓→

such that Y → V is a weak equivalence. Suppose either that X → Z is

a cofibration, or that both maps X → Y and X → V are cofibrations.

Then the map

Z ∪X Y → Z ∪X V

is a weak equivalence.

Proof In the case where X → Z is a cofibration, is a straightforward

application of the definition of left properness. Suppose that X → Y

and X → V are cofibrations. Choose a factorization of the map X → Z

intoXc→ W

f→ Z such that c is a cofibration and f is a trivial fibration.

The map

W ∪X Y → (W ∪X Y ) ∪Y V =W ∪X V

is the pushout of the weak equivalence Y → V along the cofibration

Y → W ∪X Y , so by the left properness condition, it is a weak equiv-

alence. On the other hand, the morphisms W ∪X Y → Z ∪X Y and

W ∪X V → Z ∪X V are pushouts of the weak equivalence W → Z

along the cofibrations W → W ∪X Y and W → W ∪X V respectively.

Again by left properness, these maps are weak equivalences. Hence, in

the diagram

W ∪X Y → W ∪X V

Z ∪X Y

↓

Z ∪X V

↓→

the top arrow and both vertical arrows are weak equivalences; hence the

bottom arrow is a weak equivalence as desired.


Corollary 9.5.2 Suppose M is a left proper model category and sup-

pose given a diagram

X ← Y → Z

A

↓← B

↓→ C

↓

such that the vertical arrows are weak equivalences, and the left horizon-

tal arrows are cofibrations. Then the map

X ∪Y Z → A ∪B C


Proof Applying the previous lemma on both sides gives the required

statement in the case Y = B. In general let A′ := X ∪Y B. The map

X → A′ is a weak equivalence by left properness, so A′ → A is a weak

equivalence by 3 for 2. The map B → A′ is a cofibration. Applying the

case where the middle map is an isomorphism gives the statement that

A′ ∪B C → A ∪B C

is a weak equivalence. That case, or really the previous lemma, also

implies that the map

X ∪Y Z → X ∪Y C = (X ∪Y B) ∪B C = A′ ∪B C

is a weak equivalence. Putting these together gives the required state-

ment.

We sometimes need an analogous invariance property for transfinite

compositions.

Proposition 9.5.3 Suppose M is a tractable left proper model cate-

gory. Then transfinite compositions of cofibrations are invariant under

homotopy; that is, if Xnn<β and Ynn<β are continuous sequences

indexed by an ordinal β, with cofibrant transition maps, and if we have

a map of sequences given by a compatible collection of weak equivalences

Xngn→ Yn, the induced map colimn<βXn

gβ→ colimn<βYn is a weak

equivalence.

Proof The proof is by induction on the ordinal β; we may assume it is

known for all sequences indexed by ordinals α < β. Suppose given contin-

uous sequences Xnn<β and Ynn<β with cofibrant transition maps,


and Xngn→ Yn compatible with the transition maps of the sequences.

Put Zn := Xn ∪X0 Y0. By left properness, the maps Xn → Zn are weak

equivalences, hence by 3 for 2 the same holds for the maps Zn → Yninduced by the universal property of the pushout. Furthermore, the map

X0 → colimn<βXn is a cofibration, so again by left properness the map

colimn<βXn → colimn<βZn = (colimn<βXn) ∪X0 Y0


Choose inductively Wn fitting into a diagram

Z0 → Z1 → · · · → Zn → · · ·

W0

wwwwwwwwww→ W1

i1

↓→ · · · → Wn

in

↓→ · · ·

Y0

wwwwwwwwww→ Y1

p1

↓→ · · · → Yn

pn

↓→ · · ·

such that in are trivial cofibrations and pn are trivial fibrations, for any

non-limiting ordinal n < β. At a limit ordinal assume W· is continuous,

that is Wn = limj<nWj . Note that the map Zn → Wn will still satisfy

the left lifting property against any fibration, so Zn → Wn is still a

trivial cofibration when n is a limit ordinal, and similarly once we are

done the map colimn<βZn → colimn<βWn will be a trivial cofibration.

On the other hand, for a limiting ordinal n < β the induction hypothesis

implies that the map pn is still a weak equivalence. We may also choose

W· to have cofibrant transition maps, indeed given the choice up to n we

apply the factorization property to the map Wn ∪Zn Zn+1 → Yn, this

map is a weak equivalence by left properness, so it factors as a trivial

cofibration followed by a trivial fibration.

The trivial fibration property at successor ordinals, and continuity

at limit ordinals, plus the fact that the transition maps of Y· are cofi-

brations, allow us to choose a sequence of maps Ynsn→ Wn which

are sections of pn and commute with the transition maps. The map

pβ : colimn<βWn → colimn<βYn therefore admits a section sβ given by

the colimit of the sn. This shows that pβ admits a right inverse.

We have now proven that our original map gβ admits a right inverse

in the homotopy category. However, sβ comes from a system satisfying

9.6 Quillen adjunctions 175

the same hypotheses, so sβ also admits a right inverse in the homotopy

category; but it has by construction a left inverse, so sβ is a weak equiv-

alence; then its left inverse pβ is a weak equivalence which implies that

gβ is one too.

9.6 Quillen adjunctions

Suppose M and N are model categories. A Quillen adjunction from M

to N is an adjoint pair of functors L : M → N and R : N → M ,

with L left adjoint and R right adjoint, such that:

(QA1)—L sends cofibrations to cofibrations and trivial cofibrations to

trivial cofibrations;

(QA2)—R sends fibrations to fibrations and trivial fibrations to trivial

fibrations.

Either one of these conditions implies the other, by the adjunction

formula.

If L : M → N is a functor admitting a right adjoint R such that

(L,R) is a Quillen adjunction, we say that L is a left Quillen functor,

similarly the right adjoint R is a right Quillen functor. We say that L

or R or (L,R) is a Quillen equivalence if a map L(x) → y is a weak

equivalence if and only if the corresponding map x → R(y) is a weak

equivalence.

9.7 The Kan-Quillen model category of simplicial

sets

The most important model category is the category of simplicial sets

with the structure of Quillen model category originally defined by Kan.

We denote this model category by K . As a category, it is the category

of functors ∆o → SetU where U denotes our main chosen universe. The

cofibrations are just the monomorphisms. The fibrations are the maps

satisfying the Kan lifting condition for all horns over standard simplices.

A modern proof that these classes of maps provide a model structure, is

given in [103].

For K there are particularly good generating sets. Let h(n) denote the

standard simplex of dimension n, which as a diagram ∆o → SetU is just

the functor represented by [n]. Let ∂h(n) denote the boundary, defined

by the condition that (∂h(n))k is the subset of simplices in h(n)k which


factor through some principal face h(n − 1) ⊂ h(n). The cofibrations

are generated by the inclusions of boundaries ∂h(n) → h(n). The k-th

horn ∂〈k〉h(n) is the subobject spanned by all principal faces except

the k-th one. The trivial cofibrations are generated by the inclusions

∂〈k〉h(n) → h(n).

9.8 Model structures on diagram categories

Suppose Φ is a small category, and M is a combinatorial model category.

Recall that Func(Φ,M ) is the category of functors Φ →M . There are

two main types of model structure on Func(Φ,M ) usually known as

the projective and injective model structures. In both model structures,

the weak equivalences are defined to be the objectwise ones, in other

words A → B is a weak equivalence if and only if A(x) → B(x) is a

weak equivalence for each x ∈ Φ.

In the projective structure, the fibrations are defined as the objectwise

ones. The cofibrations, on the other hand, will be generated by the ix,!(f)

where ix : x → Φ is inclusion of a single object, and f runs through

a generating set of cofibrations for M .

Dually, in the injective structure the cofibrations are defined as the

objectwise ones, but the fibrations don’t have an easy description.

Theorem 9.8.1 Suppose Φ is a small category, and M is a combinato-

rial model category. Then Func(Φ,M ) has two structures of combinato-

rial model category, the projective and the injective ones, with classes of

morphisms described above. If M is left proper then so are the injective

and projective diagram structures.

See [16], [153, Proposition A.2.8.2] for the general case. Left proper-

ness may be verified levelwise. The main difficulty is in the construction

of the injective model structure, to get a generating set for the cofibra-

tions. This is done using Lurie’s theorem which was recalled as 8.9.2

in the preceding chapter. Of course, these model structures have a long

history going back to Bousfield-Kan and others, more recently consid-

ered by Hirschhorn [116], Smith (see Beke [27], Dugger [84]), Blander

[41], Barwick [16]. For most special cases including many of the cases of

interest to us, the reader may refer to any of a number of these references.

Barwick generalises this result to the relative case [16], using a gen-

eralization of Lurie’s theorem (which we have stated above as Theorem

9.8 Model structures on diagram categories 177

8.9.2). Recall that a left Quillen presheaf is a presheaf of model cate-

gories over a base category, such that the transition functors are left

Quillen functors. The category of sections is the category of sections of

the associated fibered category.

Theorem 9.8.2 The category of sections of a left Quillen presheaf

whose values are combinatorial model categories, has an “injective” and

a “projective” combinatorial model structure.

In the case we need which corresponds to diagrams in a constant

combinatorial model category this was given by Lurie in [153]. See [16]

for the proof in general, which for the injective structure uses Lurie’s

techniques as discussed in Theorems 8.9.1 and 8.9.2 above. The notion

of left Quillen presheaf will not be used below, the above statement was

given for informational purposes.

In Chapter 13 below, we consider a variant of diagrams called unital

diagram categories. Given Φ0 ⊂ Ob(Φ) and a single model category M ,

we could define a presheaf of model categories by setting Mx := M

if x 6∈ Φ0, with Mx := ∗ for x ∈ Φ0. The unital diagram category

Func(Φ/Φ0;M ) is the category of sections of the associated fibered

category. This presheaf of categories is usually not a left Quillen presheaf

so Propositions 13.4.2 and 13.4.3 can’t be viewed as a corollary of 9.8.2,

however the techniques of proof are the same.

Another important type of model structure on certain diagram cat-

egories is the Reedy model structure, which is defined when Φ has a

structure of “Reedy category”. This lies in between the projective and

the injective structures, and indeed a closely related variant will be pro-

vide an important class of cofibrations in Chapter 15 below. Some refer-

ences for this discussion are [177] [48] [116] [87] [91] [103] [17] and [117,

Chapter 17].

A Reedy category is a category Φ provided with two subcategories

on the same object set, called the direct subcategory Φd and the inverse

subcategory Φi and a function degree from the set of objects to an ordinal

(usually ω), such that the non-identity direct maps strictly increase the

degree, the non-identity inverse maps strictly decrease the degree, and

any morphism f factors uniquely as fdf i where fd is direct and f i is

inverse. The degree of the middle object in this factorization is ≤ the

degrees of the source and target of f , and in case of equality f is either

direct or inverse (or both in which case it is the identity).

For y ∈ Φ define Latch(y) to be Φd/y minus y and Match(y) to be

y/Φi minus y.


Suppose A : Φ →M is a diagram. The latching and matching objects

at y ∈ Φ are defined to be

latch(A, y) := colimA|Latch(y),

match(A, y) := limA|Match(y).

A diagram A ∈ Func(Φ,M ) is Reedy cofibrant if the morphisms

latch(A, y) → A(y)

are cofibrations in M , and Reedy fibrant if the morphisms

A(y) → match(A, y)

are fibrations in M . A morphism Af→ B in Func(Φ,M ) is said to be

Reedy cofibrant if the maps

latch(f, y) := latch(B, y) ∪latch(A,y) A(y) → B(y)

are cofibrations, and Reedy fibrant if the maps

A(y) → match(A, y)×match(B,y) B(y) =: match(f, y)

are fibrations.

Proposition 9.8.3 The category of diagrams Func(Φ,M ) provided

with the levelwise weak equivalences, and the above classes of cofibra-

tions and fibrations, is a closed model category, fitting in the middle of

a sequence of left Quillen functors

Funcproj(Φ,M ) → FuncReedy(Φ,M ) → Funcinj(Φ,M ).

The Reedy structure is combinatorial (resp. tractable, left proper) when-

ever M is.

Proof See the references [177] [48] [116] [87] [91] [103] [17]. In partic-

ular, inheritance of the combinatorial or tractable properties is shown

by Barwick in Lemmas 3.10, 3.11 of [17]. The left properness condition

may be verified levelwise, since Reedy cofibrations are injective ones [17,

Lemma 3.1].


9.9 Pseudo-generating sets

The cofibrant generating sets for a cofibrantly generated model category

do not often have as simple and geometric a meaning as for the origi-

nal case of simplicial sets K . This problem tends to occur particularly

for the generating set for trivial cofibrations, which is often obtained by

an abstract accessibility argument leading to a set containing all triv-

ial cofibrations up to a given cardinality. In this section, we explore a

way of defining a cofibrantly generated model category using sets I and

K. The first will be the generating set for cofibrations; but the set K

will only generate the weak equivalences in a roundabout way, thus the

terminology “pseudo-generating sets”. The construction of a cofibrantly

generated model structure in Theorem 9.9.7 uses exactly the argument

where we throw in everything up to a given cardinality. Once we have the

statement of Theorem 9.9.7 we will be able to apply it in later chapters

without having to come back to this cardinality argument. The reader

hoping to avoid too much theory of model categories could therefore

skip this section and just take Theorem 9.9.7 as a “black box” for con-

structing model structures. The pseudo-generating sets I and K used

later will have some geometric meaning and hence be motivated outside

of the technicalities of model category theory.

Suppose we are given a locally presentable category M , and two sets

of morphisms I,K ⊂ Arr(M ). Here both I and K are assumed to be

small sets. Say that a morphism f : A → B is a weak equivalence if and

only if (PG) there exists a diagram

A → A′

B

f

↓→ B′

f ′

↓

such that the horizontal morphisms are in cell(K), and the morphism

f ′ is in inj(I). Note in particular that any morphism in inj(I) is a weak

equivalence.

Define the class of cofibrations to be cof(I), the trivial cofibrations to

be the cofibrations which are weak equivalences, and the fibrations to be

the morphisms satisfying right lifting with respect to trivial cofibrations.

As usual a cofibrant object means an object X such that the morphism

∅ → X is a cofibration.

Suppose the following axioms:


(PGM1)—record here the hypotheses that M is locally presentable, and

I and K are small sets of morphisms;

(PGM2)—the domains of arrows in I and K are cofibrant, and K ⊂

cof(I);

(PGM3)—the class of weak equivalences is closed under retracts;

(PGM4)—the class of weak equivalences satisfies 3 for 2;

(PGM5)—the class of trivial cofibrations is closed under pushouts;

(PGM6)—the class of trivial cofibrations is closed under transfinite com-

position.

Using these axioms, we would like to show that these classes define a

cofibrantly generated, and indeed tractable, model category structure on

M . Note that the class of trivial fibrations is defined as the intersection

of the fibrations and the weak equivalences; we don’t know a priori that

this is the same as inj(I), that will have to be proven as a consequence of

the axioms. One can say, however, that inj(I) is contained in the class of

trivial fibrations, indeed an element of inj(I) satisfies right lifting with

respect to cof (I) so it is a fibration, and from the definition (PG) it is

a weak equivalence.

One important preliminary result is Corollary 8.7.7 from the previous

chapter, saying that any morphism in cell(I) is a κ-filtered colimit of

κ-presentable cell complexes. The other main ingredient is the following

observation which is a sort of accessibility property for morphisms in

inj(I).

Lemma 9.9.1 We suppose given three regular cardinals µ < λ < κ

such that M is locally λ-presentable (hence also locally κ-presentable),

such that |I| < κ, and such that the sources and targets of arrows in I

are µ-presentable. We assume that 2µ < κ.

Suppose f : X → Y is in inj(I), and suppose it can be expressed f =

colimi∈αfi where fi : Xi → Yi are arrows between κ-presentable objects,

and α is κ-filtered. Then there is a collection of λ-filtered categories βjof size |βj | < κ, together with functors qj : βj ⊂ α, all indexed by a

κ-filtered poset j ∈ ψ, such that for any j, f(βj) := colimi∈βjfi is in

inj(I), and f = colimj∈ψf(βj).

Proof The first step is to say the following. For each i ∈ α there is


t(i) ∈ α with an arrow i → t(i), such that for any diagram

U → Xi

V

u

↓→ Yi

↓

with u ∈ I, there exists a lifting V → Xt(i) so that the two triangles in

U → Xt(i)

V

↓→

→

Yt(i)

↓

commute. Here the horizontal maps are the compositions of the previous

ones, with the transition maps Xi → Xt(i) and Yi → Yt(i).

To prove this, note that there exist liftings V → X for any dia-

gram. However, the Xi and Yi are κ-presentable, the arrows in I are

µ-presentable, and |I| < κ. Thus, the cardinality of the set of diagrams

we need to consider is < κµ ≤ κ, by Corollary 8.1.7. Since again U is

κ-presentable, there is some t ∈ α which works for each diagram; but

since α is κ-filtered we can choose a single t(i) for all the diagrams at a

given value of i. This completes the proof of the first step.

Now we can exhaust α by a family of λ-filtered subcategories βj with

|βj | < κ, indexed by j ∈ ψ where ψ is a κ-filtered partially ordered set.

We can do it in such a way that for any i ∈ βj we also have t(i) ∈ βj .

The exhaustion condition means that for any i ∈ α there is j ∈ ψ and

k ∈ βj with i → k, and it implies that f = colimj∈ψf(βj).

Now put

X(βj) := colimi∈βjXi

Y (βj) := colimi∈βjYi.

f(βj) := colimi∈βjfi

↓


We claim that f(βj) ∈ inj(I). If

U → X(βj)

V

u

↓→ Y (βj)

↓

is a diagram with u ∈ I, then since βj is λ-filtered and u is λ-small,

there exists a lifting to a diagram of the form

U → Xi

V

u

↓→ Yi

↓

for some i ∈ βj . By our choice of t(i) plus the hypothesis that t(i) ∈ βjwhenever i ∈ βj , we get a lifting V → Xt(i) → X(βj) which makes the

triangles in

U → X(βj)

V

u

↓→

→

Y (βj)

↓

commute. This shows that f(βj) ∈ inj(I).

A cofibrant replacement of an object X ∈ M is a morphism p :

X ′ → X such that p ∈ inj(I) and ∅ → X ′ is in cell(I). This exists

by the small object argument for I, and by the definition (PG) of weak

equivalence, it follows that p ∈ inj(I) is a weak equivalence.

Construct as follows a set J of trivial cofibrations with cofibrant do-

mains. Recall that the category M is locally κ-presentable. Choose a

small set N1 of representatives for the isomorphism classes of arrows

f : X → Y which are weak equivalences between κ-presentable objects.

Since the isomorphism classes of κ-presentable objects form a small set

(axiom (2) for the locally κ-presentable category M ), we can choose

N1 as a small set. For each f ∈ N1, choose a cofibrant replacement

p : X ′ → X , let f ′ : X ′ → Y be the composition f ′ = fp, and choose

a factorization f ′ = gh where h : X ′ → Z is in cell(I) and g : Z → Y

is in inj(I), in particular g is a trivial fibration. Let J be the set of all


cofibrations h obtained in this way. Using the facts that g and p are

weak equivalences, and the hypothesis that f is a weak equivalence, we

get that f ′ and then h are weak equivalences by (PGM4). Thus the el-

ements of J are trivial cofibrations with cofibrant domains. We have to

show that a map in inj(J) is a new fibration, that is that it satisfies

lifting with respect to any new trivial cofibration. Or equivalently, to

show that any new trivial cofibration is in cof (J).

Lemma 9.9.2 The elements of cof (J) are trivial cofibrations.

Proof Trivial cofibrations are closed under pushouts (PGM5) and trans-

finite compositions (PGM6), hence the elements of cell(J) are trivial

cofibrations. The class cof(J) is the closure of cell(J) under retracts;

by definition the class cof (I) of cofibrations is closed under retracts, and

by (PGM3) the class of weak equivalences is closed under retracts, so

elements of cof(J) are trivial cofibrations.

The following proposition says that the class of weak equivalences is

accessible. In this argument, we are following Barwick [16].

Proposition 9.9.3 Suppose f : X → Y is a weak equivalence. Then

f can be expressed as a κ-filtered colimit of arrows fi which are weak

equivalences between κ-presentable objects.

Proof The category Arr(M ) is locally κ-presentable, so we can write

the arrow f : X → Y as a κ-filtered colimit of arrows colimi∈αfi,

such that fi : Xi → Yi is κ-presentable in Arr(M ). In particular,

X = colimi∈αXi and Y = colimi∈αYi are expressed as κ-filtered colimits

of κ-presentable objects in M (see Lemma 8.1.3).

By the definition of weak equivalence depending on I and K, there

exists a diagram

Xa→ A

Y

f

↓b→ B

g

↓

such that a and b are in cell(K), and such that g ∈ inj(I).

Apply Corollary 8.7.7 to a and b. Thus

A = colimi∈αAi, B = colimi∈αBi

such that a and b are the colimits of systems ai : Xi → Ai and bi :


Yi → Bi, with Ai and Bi being κ-presentable, and ai, bi ∈ cell(K). For

each i we have a diagram

Xiai

→ Ai

Yi

fi

↓bi→ B = colimj∈αBj .

gi

↓

This gives a collection of maps, natural in i ∈ α,

Yi ∪Xi Ai → B

with the sources being κ-presentable. By Lemma 8.7.8 of the previous

chapter, there is a functor q : α → α together with factorizations

Yi ∪Xi Ai → Bq(i) → B

natural in i.

This gives a functor of diagrams depending on i ∈ α

Xiai→ Ai

Yq(i)

fi

↓ bq(i)→ Bq(i)

gi

↓

such that g = colimgi, a = colimai, b = colimbq(i) and ai, bq(i) ∈ cell(K).

Apply Lemma 9.9.1 to the map g ∈ inj(I); we conclude that the orig-

inal diagram may be seen as a κ-filtered colimit over j ∈ ψ of diagrams

of the form

X(βj)a(βj)→ A(βj)

Y (βj)

f(βj)

↓b(βj)→ B(βj)

g(βj)

↓

where X(βj) = colimi∈βjXi etc., and such that the maps g(βj) are in

inj(I). The maps a(βj) and b(βj) are still in cell(K), so this shows that

f(βj) : X(βj) → Y (βj) are weak equivalences. The objects X(βj) and

Y (βj) are κ-presentable, and f = colimj∈ψf(βj), which completes the

proof of the proposition.


Theorem 9.9.4 Our set of arrows J generates the trivial cofibrations,

in other words the class cof(J) equals the class of all trivial cofibrations.

In particular, inj(J) equals the class of fibrations.

Proof Suppose we are given a trivial cofibration f : X → Y . For now

suppose also that f is in cell(I).

Choose a cellular expression for f , that is a sequence Xnn≤β indexed

by an ordinal β with X0 = X , Xβ = Y , Xm = colimn<mXm when m is

a limit ordinal, and Xn → Xn+1 obtained by pushout along an element

of I.

We will choose a sequence of diagrams Xn → Zn → Y , where

Znn≤β is a transfinite sequence, such that X → Zn is in cell(J)

in particular a new trivial cofibration; Zn → Zn+1 is pushout along

an element of J , and again Zm = colimn<mZn when m is a limit ordi-

nal. The transition maps in the sequence Znn≤β are supposed to be

compatible with those of the sequence Xnn≤β and with the maps to

Y .

For the induction step at a limit ordinal m, just let Zm be the colimit

of the Zn for n < m.

Suppose n is a successor ordinal with the Zi for i ≤ n− 1 given, and

we want to choose Zn. Consider the map wn : Un → Vn in I such that

Xn−1 → Xn is pushout along wn; we have a commutative diagram

Un → Xn−1 → Zn−1

Vn

↓→ Xn

↓→ Y

↓

where the first square is a pushout. The map X → Zn−1 is a weak equiv-

alence by the inductive hypothesis, and X → Y is a weak equivalence by

assumption, so by (PGM4), the map Zn−1 → Y is a weak equivalence.

On the other hand, wn : Un → Vn is in I, so by the choice of κ it is

κ-presentable in Arr(M ). By Proposition 9.9.3 the map Zn−1 → Y is

a κ-filtered colimit of κ-presentable arrows in Arr(M ) which are new

weak equivalences—that is, elements of N1. Since wn is κ-presentable it


factors through one of them. Therefore, there exists a factorization

Una→ A

z→ Zn−1

Vn

↓b→ B

↓y→ Y

↓

with the middle vertical arrow in N1. Consider the choice of cofibrant re-

placement A′ → A used for the definition of J and, since Un is cofibrant

(PGM2), a lifting Un → A′. In particular we may replace A by A′ in the

previous diagram. Consider the choice of factorization A′ u→ C

v→ B

with u ∈ J and v ∈ inj(I), and choose a lifting c : Vn → C with vc = b

and c restricting to the given map on Un. This gives a factorization

Una→ A′ → Zn−1

Vn

↓c→ C

↓yv→ Y

↓

where the middle vertical arrow is in J . Let Zn := Zn−1 ∪A′

C be the

pushout along this arrow. ThenX → Zn is again in cell(J), in particular

it is a new trivial cofibration (Lemma 9.9.2). From the second square in

the above diagram we get a map Zn → Y compatible with the map

on Zn−1. But, given that Xn = Xn−1 ∪Un Vn we get a map Xn → Zncompatible with given map on Xn−1. This completes the inductive step,

giving the construction of the sequence Znn≤β as desired.

Letting Z := Zβ we get a map g : X → Z in cell(J) with projection

p : Z → Y and a splitting s : Y → Z, ps = 1 so that f is a retract of g

in objects under X . This shows f ∈ cof(J).

We have shown that a map f in cell(I) which is a new weak equiv-

alence, is in cof(J). Suppose f ∈ cof (I) is a weak equivalence. Choose

a factorization f = gh with Xh→ V

g→ Y where h ∈ cell(I) and

g ∈ inj(I). Then g is a weak equivalence from the definition (PG), so h

is a weak equivalence (PGM4). We have shown above that h ∈ cof(J).

On the other hand, since f ∈ cof(I) it satisfies lifting with respect to h

so there is a section s : Y → Z making f into a retract of h in objects

under X . In general the cofibrations for a set of arrows forms a class

closed under retracts, and f is a retract of h ∈ cof(J) so f ∈ cof(J).


This shows that cof (J) equals the class of trivial cofibrations. Given

a map g ∈ inj(J) it satisfies lifting with respect to cell(J), and any

trivial cofibration is a retract of something in cell(J). It follows that g

satisfies lifting with respect to any trivial cofibration, so g is a fibration.

Conversely a fibration is in inj(J) as follows from Lemma 9.9.2.

Corollary 9.9.5 Any map f : X → Y in M factors as f = gh with

Xh→ Z

g→ Y where h is a trivial cofibration (which can be assumed

even in cell(J)) and g is a fibration.

Proof This is just the factorization into h ∈ cell(J) and g ∈ inj(J)

for the set J , given by the small object argument. Note that h is a new

trivial cofibration by Lemma 9.9.2, and g is a new fibration by Theorem

9.9.4.

Lemma 9.9.6 The class of trivial fibrations is equal to inj(I).

Proof An element of inj(I) satisfies lifting with respect to all cofibra-

tions, in particular with respect to trivial ones, so inj(I) is contained in

the class of fibrations. It is contained in the class of weak equivalences

by the definition (PG), which shows that it is contained in the class of

trivial fibrations.

Suppose f : X → Y is a trivial fibration. Applying Theorem 9.9.4,

this means that f ∈ inj(J) and f is a weak equivalence. Use the small

object argument to choose a factorization of f as the composition of

Xg→ Z

p→ Y,

such that g ∈ cell(I) and p is in inj(I). It follows from the definition

(PG) that p is a weak equivalence. By 3 for 2, the map g is also a weak

equivalence, so it is a trivial cofibration. Thus, the condition that f be

a fibration implies that it satisfies right lifting with respect to g. Use

this lifting property on the square with the identity 1X along the top,

f on the right, g on the left and p on the bottom: hence there is a map

t : Z → X such that tg = 1X and ft = p. This presents f as a retract

of p, but p ∈ inj(I) and inj(I) is closed under retracts (see Lemma 8.8.2

of the previous chapter), so f ∈ inj(I).

We can now put together everything above to obtain a model category

structure on M . This follows Smith’s recognition theorem, as exposed in

[27], [16]. For the reader’s convenience we repeat the definition of weak

equivalence (PG) and the axioms equivalent to (PGM1)–(PGM6). The


present statement sums up the main result from the first two chapters

which will be used later on.

Theorem 9.9.7 Suppose M is a locally presentable category, and I ⊂

Arr(M ) and K ⊂ cof (I) are sets of morphisms. Say that a morphism

f : A → B is a weak equivalence if and only if there exists a diagram

A → A′

B

f

↓→ B′

f ′

↓

such that the horizontal morphisms are in cell(K), and f ′ is in inj(I).

Define the class of cofibrations to be cof(I). Define the trivial cofibra-

tions to be the cofibrations which are weak equivalences, and the fibra-

tions to be the morphisms satisfying right lifting with respect to trivial

cofibrations. Suppose:

—the domains of arrows in I and K are cofibrant;

—the class of weak equivalences contains inj(I), is closed under retracts,

and satisfies 3 for 2;

—the class of trivial cofibrations is closed under pushouts and transfinite

composition.

Then the class of trivial fibrations is exactly inj(I), and M with the given

classes is a cofibrantly generated and indeed tractable model category.

Proof Note that the class of weak equivalences is the one defined by

condition (PG) above, and the hypotheses stated in the theorem are

equivalent to the system of axioms (PGM1)–(PGM6). Let J be the set

of morphisms given for Theorem 9.9.4, which says that inj(J) is the class

of fibrations and cof(J) is the class of trivial cofibrations. By Lemma

9.9.6, inj(I) is the class of trivial fibrations.

Verify first the axioms for a closed model category.

Axiom (CM1) comes from the fact that M is locally presentable.

Axiom (CM2) is a hypothesis.

Axiom (CM3) for weak equivalences is a hypothesis. In a locally pre-

sentable category, for a given subset of morphisms I, the class cof(I) is

closed under retracts. This gives (CM3) for the cofibrations. The class of

fibrations is defined to be the class of morphisms which satisfy the right

lifting property with respect to the trivial cofibrations. By Lemma 8.8.2

of the previous chapter, the class of fibrations is closed under retracts to

give (CM3).


The fibrations are defined to be the maps satisfying the right lifting

property with respect to trivial cofibrations. Therefore, the trivial cofi-

brations satisfy the left lifting property with respect to any fibrations.

This is one half of (CM4). For the other half, Lemma 9.9.6 says that the

class of trivial fibrations is equal to inj(I), but the class of cofibrations

is equal to cof (I) so the cofibrations satisfy the left lifting property with

respect to trivial fibrations.

For (CM5), suppose f : X → Y is any morphism. It can be factored

by the small object argument in two ways, as

Xg→ Z

p→ Y

with g ∈ cell(J) ⊂ cof (J) and p ∈ inj(J), or as

Xg′

→ Z ′ p′

→ Y

with g′ ∈ cell(I) ⊂ cof(I) and p′ ∈ inj(I). In the first, g is a trivial

cofibration and p is a fibration by Theorem 9.9.4; in the second g′ is

a cofibration in view of the definition of cofibrations, and p′ is a triv-

ial fibration by Lemma 9.9.6. This proves the two parts of (CM5) and

completes the proof of the Quillen model structure.

Next we note that the model structure is cofibrantly generated, indeed

we have exhibited the required generating sets I and J . Axiom (CG1) is

automatic since M is locally presentable (all objects are small); axiom

(CG2a) is the definition of cofibrations and (CG2b) is Lemma 9.9.6; and

axioms (CG3a) and (CG3b) are given by Theorem 9.9.4.

As M is locally presentable, the model structure is combinatorial.

Furthermore it is tractable: indeed we have required in (PGM2) that

the domains of arrows in I are cofibrant, and the arrows in J have

cofibrant domains by construction.

It should perhaps be stressed that the main work was done in the

previous chapter in the discussion of cell complexes and the proof of

Theorems 8.7.6 and 8.9.1, then used in the proof of Theorem 9.9.4 above.

The advantage of the statement of Theorem 9.9.7 is that it makes no

reference to accessibility (other than the hypothesis that M be locally

presentable), so when we apply it later on we don’t need to do any more

work with cardinals and presentability.

We can get some further information on trivial cofibrations, fibrant

objects, and fibrations between fibrant objects. This result will, as it is

applied successively in later chapters, eventually turn into the version


for constant object set of Bergner’s result characterizing fibrant Segal

categories [36].

Proposition 9.9.8 In the situation of Theorem 9.9.7, suppose Xf→ Y

is a cofibration (i.e. in cof (I)). Then f is a trivial cofibration if and only

if there exists a diagram

Xa→ A

Y

f

↓b→ B

s

↑

g

↓

such that ga = bf , sbf = a, gs = 1B, and a, b ∈ cell(K).

An object U ∈ M is fibrant if and only if U ∈ inj(K). If this is the

case, then a morphism Wp→ U is a fibration if and only if p ∈ inj(K).

Proof If f is a new trivial cofibration, then it fits into a diagram with

a two maps a, b ∈ cell(K) and a morphism g ∈ inj(I) by the definition

of weak equivalences (PG). Since bf is cofibrant, there exists a lifting s

with gs = 1B and sbf = a. This is a diagram of the required form.

Suppose f is a cofibration such that there exists a diagram as above

(but in this case g is no longer assumed in inj(I)). Then bf is a retract

of a which is a weak equivalence. By closure of weak equivalences under

retracts and then 3 for 2, it follows that f is a new weak equivalence

hence a trivial cofibration. This proves the first statement.

If U is fibrant then it is clearly K-injective since the elements of K

are trivial cofibrations. Suppose U ∈ inj(K). If Xf→ Y is a trivial

cofibration, there exists a diagram as in the first part. For any map

X → U we can extend it to a map A → U and composing with sb gives

the required extension to Y → A. This shows that U is fibrant.

Suppose now that U is fibrant andWp→ U is a map. If p is a fibration

then clearly it is in inj(K). On the other hand suppose p ∈ inj(K), and

Xw→ W

Y

f

↓u→ U

p

↓

is a diagram with f being a trivial cofibration. Choose a diagram as in


the first part of the proposition for f . Since U is assumed fibrant, we

can extend the bottom map to a map Bt→ U with tb = u and which

composes with g to give a diagram

Xw→ W

A

a

↓tg→ U.

p

↓

Now a ∈ cell(K) and p ∈ inj(K) so there is a lifting Ar→ W with

ra = w and pr = tg. This gives a map Yrsb→ W such that rsbf = ra = w

and prsb = tgsb = tb = u, which is the required right lifting property

showing that p is a fibration.

10

Cartesian model categories

The notion of cartesian model category plays two important roles. First

of all, the Segal conditions involve direct products, so it is important that

they behave well homotopically. The second application is the fact that

a cartesian model category P leads to a P-enriched category obtained

by looking at its fibrant and cofibrant objects and using the internal

Hom to define morphism objects. So, the cartesian condition is one

of the main hypotheses but also one of the main properties which we

would like to prove for our construction of a model category PC(M )

of M -enriched precategories. This compatibility with products will be

shown in Chapter 19, furthermore it will provide a useful trick to help

establishing the model structure.

By “cartesian model category”, we mean a symmetric monoidal model

category in the sense of Hovey [120] for the monoidal operation given

by direct product. We add a condition about commutation of direct

products with colimits, in order to be able to get an internal Hom.

Recall that ∅ denotes the initial object and ∗ the coinitial object of

M .

Definition 10.0.9 Say that a combinatorial model category M is

cartesian if:

(DCL)—the direct product preserves colimits: if Aii∈α and Bjj∈βare diagrams, then

colimα×βAi ×Bj = (colimαAi)× (colimβBj);

(AST)—the map ∅ → ∗ is cofibrant;

(PROD)—for any cofibrations A → B and C → D the map

A×D ∪A×C B × C → B ×D (10.0.1)


Cartesian model categories 193

is a cofibration; if in addition at least one of A → B or C → D is a

trivial cofibration then (10.0.1) is a trivial cofibration.

See the definition of monoidal model category given for example in

[119] [188]. Our first axiom says that the unit object for the direct prod-

uct is cofibrant, which is stronger than the unit axiom of [188]. Note that

since M is combinatorial, the factorizations can be chosen functorially

so as to approach Hovey’s definition.

We record now some first consequences of this definition.

Lemma 10.0.10 Condition (DCL) implies that for any object X the

natural map ∅ → X×∅ is an isomorphism. In turn this implies that the

object ∅ is empty: if X → ∅ is any morphism, then X ∼= ∅.

Proof Note that ∅ is the colimit of the empty diagram, that is ∅ =

colim∅catF where ∅cat is the empty category and F : ∅cat →M is the

unique functor. Compatibility of products and colimits (DCL) therefore

implies that for any X = colim1X (here 1 is the one-object category)

we have

X × ∅ = colim1X × colim∅catF = colim1×∅cat(X ⊠ F ) = ∅

the last equality coming from the fact that 1× ∅cat = ∅cat is again the

empty category.

Suppose now that f : X → ∅ is a morphism. Letting e : ∅ → X denote

the unique morphism we get fe = 1∅. On the other hand, the morphism

(1X , f) : X → X × ∅ factors through (e, 1∅) : ∅∼=→ X × ∅. Projecting

to the second factor shows that the factorization map is f : X → ∅ and

projecting to the first factor then shows that 1X = ef . Thus f is an

isomorphism inverse to e.

Lemma 10.0.11 If M satisfies (PROD) and (DCL) then, for any

pair of weak equivalences Af→ B and C

g→ D, the resulting product

map A× C(f,g)→ B ×D is a weak equivalence.

Proof Suppose first that f and g are trivial cofibrations between cofi-

brant objects. Apply (PROD) to f and ∅ → C, then to ∅ → B and g,

and use (DCL) via the result of Lemma 10.0.10. These give that both

maps

A× C → B × C → B ×D

are trivial cofibrations; hence their composition is a weak equivalence.

Suppose f is a cofibration between cofibrant objects, and C is any

194 Cartesian model categories

object. Choose a replacement C′ p→ C where C′ is cofibrant and p is

a trivial fibration. By the first paragraph, A × C′ → B × C′ is a weak

equivalence. On the other hand, A×C′ → A×C and B×C′ → B×C

are trivial fibrations as can be seen directly from the lifting property.

Writing a commutative square and using 3 for 2 we get that the map

A× C → B × C is a weak equivalence.

Suppose now that f is an arbitrary weak equivalence and C is any

object. Choose a cofibrant replacement A′ → A, then complete to a

square

A′ → B′

A

↓→ B

↓

such that B′ → B is a cofibrant replacement, and A′ → B′ is a trivial

cofibration. The vertical maps are trivial fibrations so their products

with C remain trivial fibrations, and by the previous paragraph the

map A′ × C → B′ × C is a weak equivalence. By 3 for 2, the map

A× C → B × C is a weak equivalence.

Similarly B × C → B ×D is a weak equivalence, and composing we

get the statement of the lemma.

Lemma 10.0.12 Suppose M is a cartesian model category. Then if A

and B are cofibrant, so is A×B. If A → B and C → D are cofibrations

(resp. trivial cofibrations) between cofibrant objects then A×C → B×D

is a cofibration (resp. trivial cofibration).

Proof Follow the argument in the first paragraph of the proof of the

previous lemma.

Condition (DCL) holds for a wide variety of categories, notably any

presheaf category.

Lemma 10.0.13 If M = Presh(Φ) is the category of presheaves of

sets over a small category Φ, then it satisfies condition (DCL).

Proof Products and colimits are computed objectwise, and these prop-

erties hold in Set.


10.1 Internal Hom

uppose P is a tractable left proper cartesian model category. In practice,

P will be the model category PC(M ) of M -enriched precategories

which we are going to construct, starting with a tractable left cartesian

model category M . This is why we use the notation P rather than M

in the present discussion.

The underlying category of P is locally presentable, so commuta-

tion of direct product with colimits yields an internal Hom (Proposi-

tion 8.6.1). For any A,B this is an object Hom(A,B) together with a

map Hom(A,B) × A → B such that for any E ∈ M , to give a map

E → Hom(A,B) is the same as to give a map E ×A → B.

The cartesian condition is designed exactly so that the internal Hom

will be compatible with the model structure.

Theorem 10.1.1 Suppose M is a tractable cartesian combinatorial

model category. Then:

(a)—the internal Hom(A,B) exists for any A,B ∈M , and takes pushouts

in the first variable or fiber products in the second variable, to fiber prod-

ucts. For example, given morphisms A → B and A → C and any D,

we have

Hom(B ∪A C,D) = Hom(B,D)×Hom(A,D) Hom(C,D).

(b)—if A is cofibrant and B is fibrant, then Hom(A,B) is fibrant;

(c)—if A′ → A is a cofibration (resp. trivial cofibration) and B is fibrant,

then the induced map Hom(A,B) → Hom(A′, B) is a fibration (resp.

trivial fibration);

(d)—if A is cofibrant and B → B′ is a fibration (resp. trivial fibration),

then the induced map Hom(A,B) → Hom(A,B′) is a fibration (resp.

trivial fibration);

(e)—if A′ → A (resp. B → B′) is a weak equivalence between cofibrant

(resp. fibrant) objects then the induced map Hom(A,B) → Hom(A′, B′)


Proof Part (a) comes from Proposition 8.6.1, and the fiber product

formulae come from the adjunction definition of Hom and the fact that

direct product preserves colimits.

For (c), suppose Af→ A′ is a trivial cofibration and B is fibrant. Then

196 Cartesian model categories

for any cofibration Eh→ F , the map

A× F ∪A×E A′ × E → A′ × F

is a trivial cofibration. Hence B satisfies the right lifting property with

respect to it. This translates, by the adjunction property of Hom, to the

right lifting property for

f∗ : Hom(A′, B) → Hom(A,B)

along h. This shows that f∗ is a trivial fibration. Similarly if f was a

cofibration then f∗ is a fibration.

For (d), suppose A is cofibrant and Bf→ B′ is a fibration; then if

Eh→ F is a trivial cofibration, the product A×E

1A×h→ A×F is also a

trivial cofibration. Since f is fibrant, it satisfies right lifting with respect

to this product map 1A × h, which is equivalent to the right lifting

property for Hom(A, f) with respect to h by the adjunction property

of Hom. Thus Hom(A,B) → Hom(A,B′) is fibrant. Applied to B′ = ∗

this says that if B is fibrant and A cofibrant then Hom(A,B) is fibrant,

giving (b). For the other part of (d), note that similarly a trivial fibration

is tranformed to a trivial fibration.

For (e), any weak equivalence between cofibrant objects Af→ A′ can

be decomposed as f = pi where i is a trivial cofibration from A to some

A′ and p is a trivial fibration from A′′ to A′; in turn p admits a section

s : A′ → A′′ with ps = 1A′ . Now Hom transforms (contravariantly) i to

i∗ which is a trivial fibration, in particular invertible in the homotopy

category. Thus, in ho(P) we have

ho(s∗)ho(i∗)−1ho(f∗) = ho(s∗)ho(i∗)−1ho(i∗)ho(p∗) = ho(s∗)ho(p∗) = ho((ps)∗) = 1

which says that ho(f∗) admits a left inverse. On the other hand, s is also

a weak equivalence between cofibrant objects so ho(s∗) also admits a left

inverse, whereas from the above formula it admits a right inverse too.

Thus ho(s∗) is invertible, which then implies that ho(f∗) is invertible

and f∗ is a weak equivalence. This was just a contravariant version of

the standard argument which appears elsewhere.

If A is cofibrant and Eh→ F is a cofibration then A×E → A× F is

again a cofibration; it follows as usual from the adjunction property that

a trivial fibrationB → B′ induces a trivial fibrationHom(A,B) → Hom(A,B′),

and then repeating an argument similar to the previous one (but co-

variantly this time) yields that if B → B′ is a weak equivalence be-

10.2 The enriched category associated to a cartesian model category197

tween fibrant objects, then Hom(A,B) → Hom(A,B′) is a weak equiv-

alence.

10.2 The enriched category associated to a cartesian

model category

Keep the a tractable left proper cartesian model category P. Then we

obtain a P-enriched category of cofibrant and fibrant objects, denoted

Enr(P) defined as follows. It is in the next higher universe level. The

object class of Enr(P) is defined to be Ob(Pcf ), the class of cofibrant

and fibrant objects of P. For any two such objects A and B, put

Enr(P)(A,B) := Hom(A,B) ∈P.

The structural morphisms for the enriched category structure come from

the standard morphisms for the internal Hom. We get a structure of P-

enriched category.

Although Enr(P) is strictly associative as a P-enriched category, we

can also consider it as a weakly P-enriched category, that is

Enr(P) ∈ PC(P)

in the notation of Chapter 12.

An early example using an internalHom to obtain a higher categorical

structure was the notion of “enhanced triangulated category” of Bondal

and Kapranov [43].

11

Direct left Bousfield localization

Suppose M is a model category and K a subset of morphisms. A left

Bousfield localization is a left Quillen functor M → N sending ele-

ments of K to weak equivalences, universal for this property, and fur-

thermore which induces an isomorphism of underlying categories and

an isomorphism of classes of cofibrations. If it exists, it is unique up to

isomorphism.

It is pretty well-known that the left Bousfield localization exists when-

ever M is a left proper combinatorial model category. We refer to the

references and particularly Hirschhorn [116] and Barwick [16] (see also

Rosicky-Tholen [181]) for this existence theorem and for some of the

main characterizations, statements, details and proofs concerning this

notion in general. Recall that the general definition of K-local objects

and the localization functor depend on the notion of simplicial map-

ping spaces. Of course, simplicial mapping spaces are exactly the kind

of thing we are looking at in the present book, but to start with these as

basic building blocks would stretch the notion of ‘bootstrapping’ pretty

far. Therefore, in the present chapter, we consider a special case of left

Bousfield localization in which everything is much more explicit.

11.1 Projection to a subcategory of local objects

Start with a left proper tractable model category (M , I, J), that is a

left proper cofibrantly generated model category such that M is locally

κ-presentable for some regular cardinal κ, and the domains of arrows in

I and J are cofibrant.

Suppose we are given a subclass of objects considered as a full subcat-

egory R ⊂ M , and a subset of morphisms K ⊂ Mor(M ). We assume



that:

(A1)—K is a small subset;

(A2)—J ⊂ K;

(A3)—K ⊂ cof(I) and the domains of arrows in K are cofibrant;

(A4)—if X ∈ R and X ∼= Y in ho(M ) then Y ∈ R; and

(A5)—inj(K) ⊂ R.

Say that (R,K) is directly localizing if in addition to the above condi-

tions:

(A6)—for all X ∈ R such that X is fibrant (i.e. J-injective), and for any

X → Y which is a pushout by an element of K, there exists Y → Z in

cell(K) such that X → Z is a weak equivalence.

Lemma 11.1.1 Under the above hypotheses, we can remove the re-

quirement that X be fibrant in the direct localizing condition: if X ∈ R

and X → Y is a pushout by an element of K then there exists Y → Z

in cell(K) such that X → Z is a weak equivalence.

Proof Choose a map X → X ′ in cell(J), in particular a trivial cofibra-

tion, such that X ′ is fibrant. By invariance of R under weak equivalences

(A3), X ′ ∈ R. Let Y ′ := Y ∪X X ′, then X ′ → Y ′ is again a pushout by

the same element of K. The above condition now applies: there is a map

Y ′ → Z in cell(K) such that X ′ → Z is a weak equivalence. Note that

X → Z is then also a weak equivalence. On the other hand, Y → Y ′

is in cell(J) hence also in cell(K) because J ⊂ K by (A2). Thus the

composition Y → Z is in cell(K). We get the desired properties.

In our main examples, R will be the subcategory of precategories

which satisfy the Segal conditions. Pelissier called this the subcategory

of “regal objects” [171]. The main step is to useK to construct a monadic

projection from M to R up to homotopy.

Let G : M →M with ηX : X → G(X) denote a K-injective replace-

ment functor, such that ηX ∈ cell(K) for all X . This exists by small

object argument for the locally presentable category M , Theorem 8.8.1.

For any X ∈ Ob(M ), since G(X) ∈ inj(K), condition (A5) implies

that G(X) ∈ R.

Our first step will be to augment the direct localizing property (A6),

from morphisms in K to morphisms in cell(K).

Proposition 11.1.2 Under the above assumptions, if X ∈ R and

X → Y is in cell(K) then there exists Y → Z also in cell(K) such

that X → Z is a weak equivalence.

200 Direct left Bousfield localization

Proof Write Y = colimiXi with X0 = X , and the colimit ranges over

an ordinal β. Suppose this is a standard presentation of a cell complex,

that is Xi → Xi+1 is a pushout by an element of K, and for a limit

ordinal i we have Xi = colimj<iXj .

We construct a system of morphisms vi : Xi → Zi (for i ∈ β) together

with morphisms gji : Zj → Zi for j ≤ i forming a transitive system,

and starting with Z0 = X0 (and v0 is the identity), such that

—each vi is an element of cell(K);

—for a limit ordinal i we have Zi = colimj<iZj;

—each Zi ∈ R;

—each Zi → Zi+1 is a weak equivalence, and an element of cell(K).

For any ordinal α ≤ β let V (α) denote the set of such compatible

collections for i < α. We have restriction maps V (α′) → V (α) for

α ≤ α′. If α is a limit ordinal then V (α) = limη<αV (η). We will show

later that V (α+ 1) → V (α) is surjective.

Assuming this for now, we show by transfinite induction on α ≤ β

that for any η ≤ α the map V (α) → V (η) is surjective. Assuming the

contrary, let α0 be the smallest ordinal ≤ β such that this is not true.

Choose an η (which we may assume < α0) such that V (α0) → V (η) is

not surjective. Either α0 is a limit ordinal, or it is a successor ordinal.

If α0 is a limit ordinal then the limit in the above expression can be

restricted to ordinals bigger than η:

V (α0) = limη≤φ<α0V (φ).

Each of the maps V (φ) → V (η) is surjective, and the transition maps

V (φ) → V (φ′) in the inverse system are surjective, so the map

limη≤φ<α0V (φ) → V (η)

is surjective, a contradiction. Suppose α0 = α + 1 is a successor or-

dinal. Then by the above claim (which we will show next) the map

V (α0) → V (α) is surjective; hence by the induction hypothesis for any

η ≤ α the map V (α0) → V (η) is surjective, and of course it also works

for η = α0, so this again gives a contradiction. We have completed the

proof of surjectivity of V (α) → V (η) for all η ≤ α ≤ β.

Since V (0) is nonempty, this shows that V (β) is nonempty, i.e. there

exists a system of the required kind.

We now have to show the claim: that V (α+1) → V (α) is surjective.

We are given data vi for all i < α and need to construct vα : Xα → Zα.


If α is a limit ordinal, put

Zα := colimi<αZi.

Let vα be the natural map from Xα∼= colimi<αXi to Zα. We claim that

this is again in cell(K), indeed a directed colimit of maps in cell(K)

is again in cell(K). The second condition holds automatically by con-

struction. For the fourth condition, note that the maps Zi → Zα are in

cell(K) because Zα is by construction a transfinite composition of the

previous maps which were in cell(K), see Lemma 8.7.2. Furthermore,

the maps in the system Zii<α are trivial cofibrations, and a directed

limit of trivial cofibrations satisfies the required lifting property against

fibrations, so it is again a trivial cofibration. Thus Zi → Zα are trivial

cofibrations, in particular they are weak equivalences. This finishes the

fourth condition, and it also gives the third condition: since Zi ∈ R we

have Zα ∈ R by invariance of R under weak equivalences.

We now treat the other case of the claim: where α is a successor

ordinal, say α = η + 1. We need to construct vη+1 : Xη+1 → Zη+1

starting from vη. Note that Xη → Xη+1 is a pushout along an element

of K. Let W be the pushout in the square

Xη → Xη+1

Zη

↓→ W.

↓

The map along the bottom is again a pushout by the same element of

K as the map along the top. By our inductive hypothesis, Zη ∈ R.

By the assumed properties (improved in Lemma 11.1.1) there is a new

Zη+1 and a map W → Zη+1 in cell(K) such that Zη → Zη+1 is a weak

equivalence. Let vη+1 be the composed map Xη+1 → Zη+1. This map

is in cell(K) because the map Xη+1 → W is in cell(K) (since it is a

pushout of the map vη which was in cell(K)); and the map W → Zη+1

is in cell(K) by construction. This gives the first required property. The

second required property doesn’t say anything because η + 1 is not a

limit ordinal; the fourth property comes from the above construction,

and as usual the third property follows from the condition Zη ∈ R and

stability of R under weak equivalences.

This finishes the proof of the proposition.


Corollary 11.1.3 Suppose f : X → Y is a morphism in cell(K),

such that X,Y ∈ R. Then f is a weak equivalence.

Proof By the proposition, there exists a morphism ϕ : Y → Z in

cell(K) such that ϕ f : X → Z is a weak equivalence. Applying the

proposition again, there exists a morphism ξ : Z → W in cell(K) such

that ξ ϕ : Y → W is a weak equivalence. Looking at the images of

everything in the homotopy category ho(M ) we find that the image of

ϕ is a map with both left and right inverses. It follows that ϕ goes to an

isomorphism in the homotopy category, hence ϕ is a weak equivalence.

By 3 for 2 we get that f is a weak equivalence.

Corollary 11.1.4 Under the above assumptions, for an object X ∈M

the following are equivalent:

(1)—X ∈ R;

(2)—ηX : X → G(X) is a weak equivalence;

(3)—for any map f : X → Y in cell(K) such that Y ∈ inj(K), f is a

weak equivalence;

(4)—there exists a map f : X → Y in cell(K) such that Y ∈ inj(K)

and f is a weak equivalence.

Proof Suppose ηX is a weak equivalence. Note thatG(X) isK-injective,

so by our hypothesis on R we haveG(X) ∈ R. Then since R is supposed

to be stable under weak equivalences, we get X ∈ R.

Suppose X ∈ R. Apply the previous Corollary 11.1.3 to the map

ηX : X → G(X) which is in cell(K), between elements of R. It says

that ηX is a weak equivalence.

We have now shown (1) ⇔ (2). It is clear that (3) → (2) → (4).

Suppose (4) with f : X → Y a weak equivalence and Y ∈ inj(K). Then

Y ∈ R by (A5) and X ∈ R by (A4)n which is (1).

This corollary says that R is determined by K. Thus, we can say

that K is directly localizing if there exists a class of objects R satisfying

properties (A1)–(A6). The class R can be assumed to be defined by

conditions (2), (3) or (4) of the corollary.

Lemma 11.1.5 Suppose given a diagram

X

Y

f

↓g→ Z,

→


with Y, Z ∈ R such that f : X → Y and gf : X → Z are in cell(K).

Then g is a weak equivalence.

Proof Let U := Y ∪X G(X), denote the two morphisms by a : Y → U

and b : G(X) → U and compose with the map ηU : U → G(U) to get

a diagram

Xf→ Y

G(X)

ηX

↓ηU b→ G(U).

ηUa

↓

Note here that ηU b is not necessarily equal to G(bηX) (this kind of

problem is the difficulty of the present proof). By hypothesis f ∈ cell(K)

so all of the maps in this square are in cell(K).

Next, put V := Z ∪Y G(U) and let c : Z → V and d : G(U) → V

denote the morphisms. Compose again with ηV to get the diagram

Yg→ Z

G(U)

ηUa

↓ηV d→ G(V ).

ηV c

↓

The map c comes by pushout from the map ηUa : Y → G(U) which is

in cell(K), so c ∈ cell(K) and ηV c ∈ cell(K). However, we don’t know

that g, d or ηV d are in cell(K).

Put these together into a big diagram of the form

Xf→ Y

g→ Z

G(X)

ηX

↓ηU b→ G(U)

ηUa

↓ηV d→ G(V ).

ηV c

↓

(11.1.1)

All of the vertical maps are in cell(K). The horizontal maps in the

square on the left are in cell(K). The composition gf along the top is

in cell(K); we would like to show the same for the composition along


the bottom. For this, note that we have a diagram

Xf→ Y

g→ Z

G(X)

ηX

↓b→ U

a

↓→ Z ∪Y U = Z ∪X G(X)

↓

G(U)

ηU

↓

→ G(U) ∪U (Z ∪Y U) = V

ηV

↓

G(V ).

ηV

↓

The horizontal map G(X) → Z ∪X G(X) is a pushout of the morphism

gf ∈ cell(K) by the morphism ηX , so it is in cell(K). The vertical map

from Z ∪X G(X) to V is a pushout along ηU , so it is in cell(K), and the

map ηV is in cell(K), so the composed vertical map Z∪XG(X) → G(V )

is in cell(K). We conclude that the map G(X) → G(V ) is in cell(K).

This is the same as the composition ηV d ηU b along the bottom of the

previous diagram (11.1.1).

This map is in cell(K) and goes between elements of R, so it is a

weak equivalence by Corollary 11.1.3. Furthermore, the map ηU b on

the bottom left of (11.1.1) is in cell(K) and goes between elements of

R, so it is a weak equivalence. We conclude by 3 for 2 that the map

G(U)ηV d→ G(V ) is a weak equivalence.

In our previous diagram (11.1.1) the vertical maps are in cell(K), and

in the Y and Z columns these maps go between elements of R, so the

center and right vertical maps are weak equivalences. In the previous

paragraph we have seen that the bottom of this rightward square is a

weak equivalence, so by 3 for 2 we conclude that g : Y → Z is a weak

equivalence. This proves the lemma.

Corollary 11.1.6 Suppose f : X → Y is a morphism in cell(K).

Then G(f) : G(X) → G(Y ) is a weak equivalence.


Proof Apply the previous lemma to the triangle

X

G(X)

ηX

↓G(f)→ G(Y ).

→

Naturality for the transformation η says that the composition G(f)ηXis the same as ηY f . We know that ηY ∈ cell(K), and f ∈ cell(K) by

hypothesis, so G(f) ηX ∈ cell(K). On the other hand, both G(X) and

G(Y ) are in R, so Lemma 11.1.5 applies to show that G(f) is a weak

equivalence.

Corollary 11.1.7 For any object X ∈M , both maps G(ηX) and ηG(X)

from G(X) to G(G(X)) are weak equivalences.

Proof The map ηX is in cell(K) so the previous corollary shows that

G(ηX) is a weak equivalence. The map ηG(X) is in cell(K) and goes

between two objects in R, so it is a weak equivalence by Corollary 11.1.3.

11.2 Weak monadic projection

It is useful to look at the subcategory R and the functor G in general

terms. One can axiomatize their properties, in a homotopy-theoretic

analogue of the discussion of Section 8.2.

Suppose M is a model category, and R ⊂M a full subcategory. We

assume that R is stable under weak equivalences, which means that

it comes from a subset of isomorphism classes in ho(M ). The aim is

to apply the general discussion of this section to the subcategory R

of the previous section; however, formally R, and the functor G which

sometimes shows up below, are not necessarily those of the previous

section.

A weak monadic projection from M to R is a functor F : M →M

together with a natural transformation ηX : X → F (X), such that:

(WPr1)—F (X) ∈ R for all X ∈M ;

(WPr2)—for any X ∈ R, ηX is a weak equivalence;

(WPr3)—for any X ∈ M , the map F (ηX) : F (X) → F (F (X)) is a

weak equivalence;

(WPr4)—if f : X → Y is a weak equivalence between cofibrant objects


then F (f) : F (X) → F (Y ) is a weak equivalence; and

(WPr5)—F (X) is cofibrant for any cofibrant X ∈M .

Note that the map in (WPr3) is different from the map ηF (X) :

F (X) → F (F (X)) which itself is a weak equivalence by (WPr1) and

(WPr2). This differentiates the weak situation from Lemma 8.2.1 for the

case of monadic projection considered in Chapter 8. Another notable de-

tail is that in condition (WPr4) the objects are required to be cofibrant;

this is done in order to make the proof of Corollary 11.3.1 work below.

Suppose (F, η) is a weak monadic projection from M to R. Let ho(R)

denote the image of R in ho(M ). Then we can construct a monadic pro-

jection (ho(F ), ho(η)) from ho(M ) to ho(R). Because of this restriction

to cofibrant objects in (WPr4), we need to compose with a cofibrant

replacement in order to define ho(F ). This process, and hence the proof

of Lemma 11.2.1 below, can be simplified if M is an injective model cat-

egory where all objects are cofibrant—and (WPr5) would be superfluous

as well.

Let P : M →M be a functor with a natural transformation ξX :

P (X) → X such that P (X) is cofibrant and ξX is a trivial fibration for

allX ∈M . Then FP is invariant under weak equivalences: if f : X → Y

is a weak equivalence, we have a commutative diagram

P (X) → P (Y )

X

↓→ Y

↓

such that three sides are weak equivalences. Therefore P (X) → P (Y )

is a weak equivalence between cofibrant objects, and by (WPr4), the

map FP (X) → FP (Y ) is again a weak equivalence. It follows that FP

descends to a functor which we denote by ho(F ) : ho(M ) → ho(M ).

For any X ∈M , consider the diagram

X ←ξX

P (X)ηP(X)→ FP (X).

It projects in ho(M ) to a diagram where the first arrow is invertible; we

can define

ho(η)X := ho(ηP (X)) ho(ξX)−1 : ho(X) → ho(FP (X)) =: ho(F )(X).

Here we denote also by ho the functor from M to ho(M ).

Lemma 11.2.1 Given a weak monadic projection (F, η) and choosing


a cofibrant replacement functor (P, ξ), the collection ho(η) defined above

is a natural transformation, and the pair (ho(F ), ho(η)) is a monadic

projection from ho(M ) to ho(R).

Proof For naturality of ho(η), suppose f : X → Y is a morphism in

M . Then we have a diagram whose vertical arrows come from f ,

X ←ξX

P (X)ηP (X)→ FP (X)

Y

↓←

ξYP (Y )

↓ηP (Y )→ FP (Y )

↓

which commutes by naturality of ξ and η. The image of this is a commu-

tative diagram in ho(M ) where the first horizontal arrows are isomor-

phisms; replacing them by their inverses and taking the outer square we

get a commutative diagram

Xho(η)X

→ ho(F )(X)

Y

↓ho(η)Y

→ ho(F )(Y )

↓

which shows naturality of ho(η) with respect to morphisms coming from

M . The same diagram implies naturality with respect to the inverse

of such a morphism, when f is a weak equivalence. These two classes

of morphisms generate the morphisms of ho(M ) so ho(η) is a natural

transformation.

It remains to check the conditions (Pr1)–(Pr3) of a monadic pro-

jection. Condition (WPr1) implies that ho(F )(X) = ho(FP (X)) is in

ho(R) for any X , giving (Pr1).

For property (Pr2), suppose X ∈ R (note that R and ho(R) have

the same objects). Then P (X) → X is a weak equivalence, so by

the invariance of R with respect to weak equivalences P (X) ∈ R,

and condition (WPr2) says that ηP (X) is a weak equivalence. Therefore

ho(η)X = ho(ηP (X)) ho(ξX)−1 is an isomorphism.

For property (Pr3), suppose X ∈ M . Recall that ho(F ) is obtained

by descending the functor FP to the homotopy category. In order to

apply this to a composition such as ho(η)X = ho(ηP (X)) ho(ξX)−1, we


use the fact that FP takes weak equivalences to weak equivalences to

say that FP (ξX) is a weak equivalence, and then

ho(F )(ho(η)X) = ho(F )(ho(ηP (X)) ho(ξX)−1

)

where the right side is defined to be ho(FP (ηP (X))) ho(FP (ξX))−1.

Consider the commutative diagram obtained by applying F to the nat-

urality diagram for ξ with respect to ηP (X):

FP (P (X))FP (ηP (X))

→ FP (FP (X))

FP (X)

F (ξP (X))

↓F (ηP (X))

→ F (FP (X))

F (ξFP (X))

↓

.

By condition (WPr3), the bottom arrow F (ηP (X)) is a weak equiva-

lence. On the other hand, P (X) is cofibrant, so ξP (X) is a weak equiva-

lence between cofibrant objects; by (WPr4), F (ξP (X)) is a weak equiv-

alence. Similarly, (WPr5) says that FP (X) is cofibrant, so F (ξFP (X))

is a weak equivalence. Three of the four sides of the square are weak

equivalences, so the fourth side FP (ηP (X)) is a weak equivalence. Hence

ho(FP (ηP (X))) is an isomorphism, therefore ho(F )(ho(η)X) is an iso-

morphism. This proves (Pr3).

Proposition 11.2.2 Suppose R ⊂ M are as above, and (F, η) and

(G,ϕ) are two weak monadic projections from M to R. Then, for any

cofibrant object X ∈ M the maps F (ϕX) : F (X) → F (G(X)) and

G(ηX) : G(X) → G(F (X)) are weak equivalences, and the diagram of

weak equivalences

F (X)F (ϕX)

→ F (G(X))

G(F (X))

ϕF (X)

↓

←G(ηX)

G(X)

ηG(X)

↑

becomes a commuting diagram of isomorphisms in the homotopy category

ho(M ).

Proof Use Proposition 8.2.2 which is the same as the present statement

but for for monadic projections. In the case where all objects of M are

cofibrant, we wouldn’t need a cofibrant replacement in the construction


for Lemma 11.2.1 above and the conclusion of the proposition follows

directly. Otherwise, we need to unwind the occurrences of the cofibrant

replacement functor in the conclusion.

According to Lemma 11.2.1, (ho(F ), ho(η)) and (ho(G), ho(ϕ)) are

monadic projections from ho(M ) to ho(R). Use the same cofibrant re-

placement (P, ξ) to define both of these. Recall that

ho(ϕ)X := ho(ϕP (X)) ho(ξX)−1

and

ho(F )(ho(ϕ)X) := ho(FP (ϕP (X))) ho(FP (ξX))−1.

Assume that X is cofibrant. Apply Proposition 8.2.2 to conclude that

ho(F )(ho(ϕ)X) is an isomorphism in ho(M ). Hence the same for ho(FP (ϕP (X))

so FP (ϕP (X) is a weak equivalence. Look at F applied to the diagram

of naturality for ξ with respect to ϕP (X)

FP (P (X))FP (ϕP (X))

→ FP (GP (X))

FP (X)

F (ξP (X))

↓F (ϕP (X))

→ F (GP (X))

F (ξGP (X))

↓

.

The vertical arrows are obtained by applying F to weak equivalences

between cofibrant objects: use condition (WPr5) for GP (X). The top

is a weak equivalence as stated above, so we conclude that the bottom

arrow F (ϕP (X)) is a weak equivalence. Now look at the diagram

FP (X)F (ϕP (X))

→ F (GP (X))

F (X)

F (ξX)

↓F (ϕX)

→ F (G(X))

FG(ξX)

↓

.

By the assumption that X is cofibrant, and by (WPr5) for both F and

G, we get that the vertical arrows are weak equivalences. The top is a

weak equivalence by the previous square diagram, so we conclude that

the bottom arrow F (ϕX) is a weak equivalence.

The other statement that G(ηX) is a weak equivalence, is obtained by

symmetry.


The commutativity of the square diagram follows from the correspond-

ing part of Proposition 8.2.2, using the same technique as above for

removing occurences of P .

11.3 New weak equivalences

We now go back to the situation of the first section, with a combinatorial

model category M with subcategory R ⊂ M and set of morphisms

K satisfying axioms (A1)–(A6), and K-injective replacement functor

(G, η).

Lemma 11.3.1 The pair (G, η) is a weak monadic projection from M

to R.

Proof For (WPr1), if X ∈M then by definition of G we have G(X) ∈

inj(K), but inj(K) ⊂ R by condition (A5).

For (WPr2), suppose X ∈ R. Then ηX is a weak equivalence by

Corollary 11.1.4.

For (WPr3), for any X ∈ M we have G(ηX) a weak equivalence by

the previous corollary.

For (WPr4), we first consider the case when f is a trivial cofibration

between cofibrant objects. Choose a factorization

Xh→ Z

g→ Y

such that h ∈ cell(J) and g ∈ inj(J). Then, since f ∈ cof (J) there is a

lifting s : Y → Z such that sf = h and gs = 1Y . Thus, f is a retract

of h in the category of objects under X . Applying the functor G we get

that G(f) is a retract of G(h) in the category of objects under G(X). On

the other hand, J ⊂ K by hypothesis (A2), so cell(J) ⊂ cell(K) and

h ∈ cell(K). By Corollary 11.1.6, G(h) is a weak equivalence. But weak

equivalences are closed under retracts, so G(f) is a weak equivalence.

This treats the case of a trivial cofibration.

Now for a general weak equivalence f : X → Y between cofibrant

objects, consider the map f ⊔ 1Y : X ∪∅ Y → Y . Choose a factorization

X ∪∅ Yh⊔s→ Z

g→ Y

such that h ⊔ s is an I-cofibration and g is a trivial fibration. Using

the fact that X and Y are cofibrant objects, and also that f is a weak

equivalence, we get that both maps h : X → Z and s : Y → Z are

trivial cofibrations between cofibrant objects. Hence G(h) and G(s) are

11.3 New weak equivalences 211

weak equivalences. But G(g), being a left inverse to G(s), is therefore a

weak equivalence, so G(f) = G(g)G(h) is a weak equivalence as desired

to show (WPr4).

To show (WPr5), note simply that ηX : X → G(X) is in cell(K), so

if X is cofibrant then G(X) is cofibrant too.

Fix a cofibrant replacement functor and natural transformation (P, ξ).

We say that a morphism f : X → Y is a new weak equivalence if

GP (f) : GP (X) → GP (Y ) is a weak equivalence. Using the theory of

the previous section, this notion depends only on R and not on K. In

passing we also show that it doesn’t depend on P .

Lemma 11.3.2 Suppose (H,ψ) is a monadic projection from ho(M )

to ho(R). Then f is a new weak equivalence if and only if H(ho(f)) is

an isomorphism.

Proof By Lemma 11.2.1, (ho(G), ho(η)) defined using the original cofi-

brant replacement (P, ξ) is also a monadic projection from ho(M ) to

ho(R). Applying Proposition 8.2.2 to compare these, gives thatH(ho(f))

is an isomorphism if and only if ho(G)(f) is an isomorphism. In turn,

ho(G)(f) = ho(GP (f)) is an isomorphism if and only if GP (f) is a weak

equivalence.

We can choose (H,ψ) = (ho(G), ho(η)), so we can say that f is a new

weak equivalence if and only if ho(G)(ho(f)) is an isomorphism.

If (F, ϕ) is any other weak monadic projection and (P ′, ξ′) is a cofi-

brant replacement functor not necessarily the same as (P, ξ) then H =

ho′(F ) and ψ = ho′(ϕ) defined as above using (P ′, ξ′) form a monadic

projection from ho(M ) to ho(R). So we can also say that f : X → Y is

a new weak equivalance if and only if ho′(F )(ho(f)) is an isomorphism,

which is if and only if FP ′(f) is a weak equivalence.

Corollary 11.3.3 The notion of new weak equivalence depends only

on R ⊂M .

Proof The condition of the lemma clearly depends only on R.

Corollary 11.3.4 If f is a weak equivalence in the original model

structure for M , then it is a new weak equivalence.

Proof In this case ho(f) is an isomorphism, so after application of a

functor ho(G)(ho(f)) remains a weak equivalence.


Corollary 11.3.5 Suppose f : X → Y is a morphism between cofi-

brant objects. Then it is a new weak equivalence, if and only if G(f) :

G(X) → G(Y ) is an old weak equivalence. If (F, ϕ) is a different weak

monadic projection then f is a new weak equivalence if and only if F (f)


Proof If X and Y are cofibrant then ho(F )(f) is isomorphic to the

projection to ho(M ) of the morphism F (f). This applies in particular

to F = G. Conclude by using Lemma 11.3.2.

The same can be said without specifying a full functor, but just looking

at X and Y .

Corollary 11.3.6 Given a morphism f : X → Y between cofibrant

objects and a diagram

Xa→ A

Y

f

↓b→ B

p

↓

with a, b ∈ cell(K) and A,B ∈ R, the map f is a new weak equivalence

if and only if p is a weak equivalence for the original model structure.

Proof Apply G to the above diagram. Corollary 11.1.6 says that the

resulting horizontal arrows are old weak equivalences. Thus, by Corollary

11.3.5, the map f is a new weak equivalence if and only ifG(A)G(p)→ G(B)

is an old weak equivalence. However, in the naturality square for η with

respect to p

AηA→ G(A)

B

↓ηB→ G(B)

↓

the horizontal arrows are old weak equivalences, since A and B are

cofibrant objects in R (see Corollary 11.1.4). Putting these statements

together using 3 for 2 gives the proof that f is a new weak equivalence

if and only if p is an old weak equivalence.


Lemma 11.3.7 The notion of new weak equivalence is stable under

retracts, and satisfies 3 for 2.

Proof It is the pullback of the notion of isomorphism, by the monadic

projection ho(G).

11.4 Invariance properties

The transfinite consequence of the left properness condition, Proposition

9.5.3, allows us to prove invariance of the new trivial cofibrations with

respect to transfinite composition. Say that a morphism is a new trivial

cofibration if it is a cofibration and a new weak equivalence.

Lemma 11.4.1 Suppose Xnn≤β is a continuous transfinite sequence

indexed by an ordinal β, such that for any limit ordinal n we have Xm =

colimn<mXn, and for any n with n+ 1 ≤ β we have that Xn → Xn+1

is a new trivial cofibration. Then X0 → Xβ is a new trivial cofibration.

Proof Assume first that X0 and hence all the Xn are cofibrant. Choose

a sequence Znn≤β and a morphism of sequences Xn → Zn in the

following way: let Z0 := G(X0); if n + 1 ≤ β and the morphisms up

to Xn → Zn are chosen, let Zn+1 := G(Zn ∪Xn Xn+1); and if m

is a limit ordinal and the morphisms are chosen for all n < m, put

Zm := colimn<mZn. This defines the full sequence by transfinite induc-

tion. Furthermore, by induction we see that the morphisms Xn → Znare in cell(K). This is clear at n = 0 and at n + 1 if we know it at

n, by the definition. In the case where m is a limit ordinal, note that

Zm = colimn<m(Xm ∪Xn Zn) using the continuity property of X·. The

transition maps in this colimit are of the form

Xm ∪Xn Zn = Xm ∪

Xn+1 (Zn ∪Xn Xn+1)

Xm ∪Xn+1 Zn+1 = Xm ∪

Xn+1 G(Zn ∪Xn Xn+1),

↓

which is in cell(K). By definition cell(K) is closed under transfinite

composition, so the map Xm → Zm is in cell(K) as claimed. It follows

(from Corollaries 11.1.6 and 11.3.5 together using the fact that Xm are

cofibrant) that the maps Xn → Zn are new weak equivalences.

We now argue by induction that the Zn are in R and the maps


Z0 → Zn are old trivial cofibrations. If it is known for Zn then the map

Zn → Zn+1 is a new weak equivalence between fibrant objects in R, so

by Corollary 11.4.5 it is an old weak equivalence, thus Z0 → Zn is an

old weak equivalence. If m is a limit ordinal and we know the statement

for all smaller n < m, then the transition maps in the system Znn<mare old trivial cofibrations; so the map Z0 → Zm = colimn<mZn is an

old trivial cofibration (trivial cofibrations are closed under transfinite

composition as can be seen from their characterization by the lifting

property with respect to fibrations). Since Z0 ∈ R we get that Zn ∈ R

by the invariance property (A4). This completes the inductive proof for

the statement given at the start of the paragraph.

Applying it to n = β we conclude that Z0 → Zβ is a new trivial

cofibration, proving the lemma under the beginning assumption that

X0 was cofibrant.

Recall that M is left proper, which implies a left properness statement

for transfinite compositions also, Proposition 9.5.3.

In the arbitrary case, choose a sequence Ynn≤β with a morphism

of sequences consisting of old weak equivalences pn : Yn → Xn, such

that the Yn are cofibrant and the transition morphisms Yn → Yn+1 are

cofibrations. This is done by using the factorization into inj(I) cell(I)

at each stage. At a limit ordinal m given the choice for all n < m,

we set Ym := colimn<mYn. By the transfinite left properness property

Ym → Xm is an old weak equivalence. This allows us to make the

inductive choice of the sequence Y·. Then, the Yn → Yn+1 are new

trivial cofibrations by Lemma 11.3.7 and the objects Yn are cofibrant, so

the first part of the proof applies: Y0 → Yβ is a new weak equivalence.

By 11.3.7 again, X0 → Xβ is a new weak equivalence.

Using the left properness hypothesis on M we can show that mor-

phisms in cell(K) are new weak equivalences, and also improve the ear-

lier criteria by removing the conditions that the morphism goes between

cofibrant objects.

Proposition 11.4.2 A morphism f : X → Y in cell(K) is a new

weak equivalence.

Proof Corollaries 11.1.6 and 11.3.5 together immediately imply this

when X is cofibrant. However this is not sufficient in general, because

new weak equivalences are defined using a cofibrant replacement. Of

course, in case M satisfies the additional hypothesis that all objects are


cofibrant, then this part of the argument (like many others) is consider-

ably simplified.

To prove that cell(K) is contained in the new weak equivalences, in

view of the closure of new trivial cofibrations under transfinite compo-

sition (Lemma 11.4.1 above), it suffices to treat morphisms which are

pushouts along a single arrow in K. Suppose X ← U → V is a diagram

with U → V in K, then we need to show that X → Y := X ∪U V

is a new weak equivalence. The source U is assumed to be cofibrant by

Condition (A3). Choose a cofibrant replacement via a trivial fibration

P → X . The map from U lifts to a map U → P . Put Z := P ∪U V ,

then we have a cocartesian square

P → Z

X

↓

→ Y

↓

where the left vertical arrow is an old weak equivalence. By left proper-

ness of the original model structure, the right vertical arrow is also an

old weak equivalence. The top map is in cell(K) and goes between cofi-

brant objects, so as mentioned above it is a new weak equivalence. Now

3 for 2 given by Lemma 11.3.7 implies that the bottom map is a new

weak equivalence.

We can now obtain a criterion for a morphism to be a weak equiv-

alence. Notice that this condition coincides with the definition of weak

equivalences (PG) used in the construction of a model category by

pseudo-generating sets in Section 9.9.

Corollary 11.4.3 A morphism f : X → Y is a new weak equivalence

if and only if there exists a diagram

Xa→ A

Y

f

↓b→ B

g

↓

such that a and b are in cell(K) and g ∈ inj(I).

Proof If such a diagram exists, then g is an old weak equivalence hence

a new one; also a and b are new weak equivalences by Corollary 11.4.2, so


by 3 for 2 for the new weak equivalences, given in the previous corollary,

we get that f is a new weak equivalence.

Suppose f is a new weak equivalence. Fix a cofibrant replacement func-

tor and natural transformation (P, ξ). Then GP (f) : GP (X) → GP (Y )

is an old weak equivalence. The map P (X) → GP (X) is in cell(K). Let

X ′ be the pushout in the cocartesian diagram

P (X)ηP (X)→ GP (X)

X

ξX

↓u→ X ′

r

↓

and similarly let Y ′ be the pushout in the cocartesian diagram

P (Y )ηP (Y )→ GP (Y )

Y

ξY

↓v→ Y ′.

s

↓

The pushout maps u and v are in cell(K). The maps ξX and ξY are old

weak equivalences, and ηP (X) and ηP (Y ) are old cofibrations, so the left

properness hypothesis on the old model structure implies that the maps

r : GP (X) → X ′ and s : GP (Y ) → Y ′ are weak equivalences. The

pushouts fit into a commutative cube, and the condition that GP (f) :

GP (X) → GP (Y ) is an old weak equivalence implies, by 3 for 2 in the

old model structure, that the map X ′ → Y ′ is an old weak equivalence.

We can factor this map as

X ′ h→ X ′′ g

→ Y ′

such that h ∈ cell(J) and g is an old fibration; but it is an old weak

equivalence too so g ∈ inj(I). We have obtained the desired diagram

Xhu→ X ′′

Y

f

↓v→ Y ′

g

↓


with hu ∈ cell(K) because it is the composition of u ∈ cell(K) with

h ∈ cell(J) ⊂ cell(K), with v ∈ cell(K), and with g ∈ inj(I).

We have the following criterion for new trivial cofibrations, which is

the same as in Proposition 9.9.8. We repeat the proof here in order to

verify that it works in our current situation.

Corollary 11.4.4 A cofibration Xf→ Y is a new trivial cofibration if

and only if it fits into a diagram

Xa→ A

Y

f

↓b→ B

s

↑

g

↓

commutative using either arrow on the right, with gs = 1B, such that a

and b are in cell(K).

Proof The same as the first part of Proposition 9.9.8, using some things

that we already know: new weak equivalences are closed under retracts

and satisfy 3 for 2 by Lemma 11.3.7, and they contain cell(K) by Propo-

sition 11.4.2.

Corollary 11.4.5 A morphism f : X → Y such that X,Y ∈ R, is

a new weak equivalence if and only if it is a weak equivalence in the

original model structure.

Proof The “if” direction is given by Corollary 11.3.4. Suppose given a

new weak equivalence f such thatX,Y ∈ R. Then P (f) : P (X) → P (Y )

is a new weak equivalence between cofibrant objects which are again in

R (since R is invariant under weak equivalences). In the diagram

P (X) → P (Y )

GP (X)

↓

→ GP (Y )

↓

the bottom arrow is an old weak equivalence by Corollary 11.3.5. The

vertical arrows are old weak equivalences by Corollary 11.1.4, (1) → (2).

By 3 for 2 in the original model structure, the top arrow P (f) is an old

weak equivalence, hence f is also.


11.5 New fibrations

Recall that a new trivial cofibration is a cofibration (which means the

same thing in the original and new structures) and also a new weak

equivalence. Say that a morphism is a new fibration if it satisfies the

right lifting property with respect to new trivial cofibrations.

Lemma 11.5.1 The class of new trivial cofibrations is closed under

composition and retracts, and contains cof(K). In particular, trivial fi-

brations from the original model structure, which are exactly cof(J), are

also new trivial fibrations.

Proof By Lemma 11.3.7, the new weak equivalences are closed under

retracts and compositions; the same holds for the cofibrations, so the

intersection of these classes is closed under retracts and compositions.

Elements of cell(K) are cofibrations by (A3) and new weak equivalences

by 11.1.6 so they are new trivial cofibrations, and cof (K) is the closure

of cell(K) under right retracts. Finally, since J ⊂ K by assumption

(A2) we get that cof(J) ⊂ cof(K).

The reader might hope that cof(K) is equal to the class of new trivial

cofibrations; however, in general it will be smaller and the next section

below will be needed to remedy this problem.

Corollary 11.5.2 A new fibration is also a fibration in the old model

structure.

Proof A new fibration satisfies right lifting with respect to the class

of old trivial cofibrations, since that is contained in the class of new

ones.

Lemma 11.5.3 A morphism which is a new fibration and a new weak

equivalence, is a trivial fibration in the original model structure, in par-

ticular it satisfies the right lifting property with respect to cofibrations.

Proof Suppose f : X → Y is a new fibration and a new weak equiv-

alence. Choose a factorization f = gh in the original model category

structure Xh→ Z

g→ Y with h a cofibration and g a trivial fibration.

Then g is an old weak equivalence, hence it is a new weak equivalence

by Corollary 11.3.4. By 3 for 2 for the new weak equivalences Lemma

11.3.7 we conclude that h is a new weak equivalence, hence by definition

it is a new trivial cofibration. The condition that f be a new fibration

implies that it satisfies lifting with respect to h, so there is a morphism

11.5 New fibrations 219

u : Z → X such that uh = 1X and fu = g. In this way, f becomes a

retract of g in the category of objects over Y . By closure of the origi-

nal weak equivalences under retracts, f is an original weak equivalence.

Since it is also a fibration in the original structure by Corollary 11.5.2, we

conclude that it is a trivial fibration in the original model structure.

Corollary 11.5.4 The class of new trivial fibrations, defined as the

intersection of the new fibrations and the new weak equivalences, is equal

to the original class of trivial fibrations.

Proof One inclusion is given by the preceding lemma. In the other

direction, suppose f : X → Y is an original trivial fibration. Then it

satisfies lifting with respect to cofibrations, in particular with respect

to new trivial cofibrations. So it is a new fibration. It is an old weak

equivalence, hence a new one by Corollary 11.3.4. Thus it is a new trivial

fibration.

We can sum up the preceding discussion as follows.

Scholium 11.5.5 The three classes of morphisms: new weak equiva-

lences, new fibrations, and cofibrations which are the same as the old

ones, generate the notions of new trivial fibration and new trivial cofi-

bration by intersection of new fibrations and cofibrations, with new weak

equivalences. All of these classes are closed under composition and re-

tracts, and new weak equivalences satisfy 3 for 2. If

Xa→ U

Y

i

↓b→ V

p

↓

is a diagram, and if either i is a new trivial cofibration and p a new

fibration; or i a cofibration and p a new trivial fibration, then there exists

a lifting f : Y → U with fi = a and pf = b.

We can add that the original model structure provides one of the two

required factorizations: if f : X → Y is any map, then it can be factored

as

f = gh : Xh→ Z

g→ Y

where h is a cofibration and g is an old or new trivial fibration (these

being the same 11.5.4).


11.6 Pushouts by new trivial cofibrations

The class of new trivial cofibrations, defined to be the intersection of the

new weak equivalences with the cofibrations, is closed under pushout:

Lemma 11.6.1 If

Xf→ Y

Z

a

↓g→ W

b

↓

is a pushout square with f a new trivial cofibration, then g is a new

trivial cofibration.

Proof First reduce to the case when X and hence Y are cofibrant ob-

jects. Choose a cofibrant replacement p : X ′ → X with p ∈ inj(I), and

a factorization of the map from X ′ to Y to give a commutative square

X ′ f ′

→ Y ′

X

p

↓f→ Y

q

↓

such that f ′ is a cofibration and q ∈ inj(I). Note that f ′ is a new trivial

cofibration, by 3 for 2 for new weak equivalences 11.3.7.

By the hypothesis that M is left proper, the map Y ′ → X ∪X′

Y ′ is

an old weak equivalence, so by 3 for 2 the map X ∪X′

Y ′ → Y is an old

weak equivalence. We get a map

Z ∪X′

Y ′ = Z ∪X (X ∪X′

Y ′) → Z ∪X Y =W.

The maps from X to X ∪X′

Y ′ and Y are cofibrations, so Lemma 9.5.1

applies: the map Z ∪X′

Y ′ → Z ∪X Y =W is an old weak equivalence.

Assume we know the statement of the lemma for cofibrant objects; then

Z → Z ∪X′

Y ′ is a new weak equivalence. The composition of these two

maps is the same as the map g : Z → W so by 3 for 2 for new weak

equivalences 11.3.7, we get that g is a new weak equivalence.

This reduces the statement of the lemma to the case where X and Y

are cofibrant, which we now suppose. Let ηX : X → G(X) be the map in

cell(K) to G(X) ∈ inj(K). Let V := G(Y ∪X G(X)). The map Y → V


is in cell(K) and V ∈ inj(K). Taking the pushout of the commutative

square

Xf→ Y

G(X)

ηX

↓t→ V

s

↓

along the map a : X → Z gives a square

Zg→ Z ∪X Y

Z ∪X G(X)

Z ∪X ηX

↓Z ∪X t

→ Z ∪X V

Z ∪X s

↓

and this fits into a cube with the previous diagram. Since f is a new

weak equivalence between cofibrant objects, the vertical maps in the first

square are in cell(K), and the objects G(X) and V are in inj(K) ⊂ R,

the map t : G(X) → V is an old weak equivalence by Corollary 11.3.6.

It is also a cofibration, so it is an old trivial cofibration. Its pushout

Z ∪X G(X)Z∪X t→ Z ∪X V is therefore an old trivial cofibration. On the

other hand, the vertical maps in the second square, being pushouts of

the vertical maps which were in cell(K) for the first square, are also

in cell(K). In particular, these maps are in cof(K) hence new weak

equivalences by Lemma 11.5.1. The bottom map is an old, hence a new

weak equivalence. By 3 for 2, Lemma 11.3.7 the map g : Z → Z ∪X Y =

W is a new weak equivalence. This proves the lemma since g is clearly

a cofibration.

11.7 The model category structure

We can now put together everything above to obtain the new model

category structure on M , as an application of Theorem 9.9.7 in the

previous chapter, which was our version of Smith’s recognition theorem

as exposed in [16].

Theorem 11.7.1 Suppose M is a left proper tractable model category,


and (R,K) is a directly localizing system according to axioms (A1)–

(A6) at the start of the chapter. The classes of original cofibrations,

new weak equivalences, and new fibrations constructed above provide M

with a structure of closed model category, cofibrantly generated, and in-

deed combinatorial and even tractable. It is left proper. Furthermore,

this structure is the left Bousfield localization of M by the original set

of maps K.

The fibrant objects are theK-injective objects, and a morphismW → U

to a fibrant object is a fibration if and only if it is in inj(K).

Proof We will be applying the discussion of Section 9.9 in the previous

chapter, about pseudo-generating sets. We are given subsets I and K;

the cofibrations are cof(I), and by Corollary 11.4.3 the new weak equiv-

alences are exactly the class defined by the condition (PG). The trivial

cofibrations are as defined in Section 9.9, hence the fibrations too. We

verify the axioms for pseudo-generating sets.

(PGM1)—The hypothesis that M is locally presentable, is contained in

the tractability hypothesis of the present statement; I is a small set as

it is one of the generating sets for the old structure, and K is a small

set by (A1).

(PGM2)—the domains of arrows in I are cofibrant, by the assumption

that the old structure is tractable (and I is part of a tractable pair of gen-

erating sets); the domains of arrows in K are cofibrant, and K ⊂ cof(I),

by (A3);

(PGM3)—the class of weak equivalences is closed under retracts by

Lemma 11.3.7;

(PGM4)—the class of weak equivalences satisfies 3 for 2 by Lemma

11.3.7;

(PGM5)—the class of trivial cofibrations is closed under pushouts by

Lemma 11.6.1;

(PGM6)—the class of trivial cofibrations is closed under transfinite com-

position by Lemma 11.4.1.

By Theorem 9.9.7, M with these classes of morphisms is a tractable

model category.


To prove left properness, suppose given a cocartesian diagram

Xu→ U

Y

f

↓v→ V

g

↓

where f is a new weak equivalence and u is a cofibration. Consider a

fibrant replacement h : Y → Y ′ and put V ′ := Y ′ ∪Y V = Y ′ ∪X U .

Since h is a new trivial cofibration, so is the map V → V ′. Factor the

composed map hf as Xi→ X ′ p

→ Y ′ where i is a new trivial cofibration

and p is a trivial fibration. In particular p is a weak equivalence in the

old model structure. The map U → X ′ ∪X U is a trivial cofibration

by Lemma 11.6.1, and left properness of the original model structure

implies that

X ′ ∪X U → Y ′ ∪X′

(X ′ ∪X U) = V ′

is an old weak equivalence (hence a new one). Composing these we get

that U → V ′ is a new weak equivalence, and by 3 for 2 we conclude

that g is a new weak equivalence.

The resulting model category is the left Bousfield localization of M

along the subset K. The morphisms of K go to weak equivalences in the

new structure. On the other hand, by the criterion of Corollary 11.4.3,

given a model structure whose class of weak equivalences contains the old

ones plus K, that class must also contain the new weak equivalences. So

the class of new weak equivalences is the smallest one which can create a

model structure along with the old cofibrations. This says that the new

structure is the left Bousfield localization [116] [16]. Alternatively, the

argument given at the end of the proof of Theorem 11.8.1 below gives

explicitly the left Bousfield property.

The characterization of fibrant objects and of fibrations to fibrant

objects in the last paragraph, is given by Proposition 9.9.8.

The left Bousfield localization depends only on the class R and not

on the choice of subset K.

Proposition 11.7.2 Suppose (R,K) and (R,K ′) are two direct local-

izing systems for the same class of objects R in a left proper tractable

model category M . Then the two new model structures given by the pre-

vious theorem are the same.


Proof By Corollary 11.3.3 the classes of new weak equivalences are the

same, and by definition the classes of cofibrations are the same, so the

classes of fibrations are the same.

11.8 Transfer along a left Quillen functor

Suppose F : M ←→ N : E is a Quillen pair of adjoint functors between

model categories M and N , with F the left adjoint and E the right

adjoint. Suppose that M and N are both tractable and left proper.

Let I and J be cofibrant generating sets for M , and I ′ and J ′ cofibrant

generating sets for N .

Suppose that R ⊂ M is a full subcategory, and K ⊂ Arr(M ) is

a small subset of arrows. Let K ′ := F (K) ∪ J ′ ⊂ Arr(M ) be the set

consisting of J ′ plus all the F (f) for f ∈ K. Let R′ := RE−1(R) ⊂ N

be the full subcategory consisting of those objects Y ∈ N such that,

for a fibrant replacement Y → Y1 we have E(Y1) ∈ R. By condition

(A4) for R and the fact that E preserves equivalences between fibrant

objets, membership of Y in R is independent of the choice of fibrant

replacement Y1.

Theorem 11.8.1 In the above situation, suppose that (R,K) is a

direct localizing system in M . Define (R′,K ′) as above. Then (R′,K ′)

is a direct localizing system in N . The functor F is a left Quillen functor

from the left Bousfield localization of M along K, to the left Bousfield

localization of N along K ′.

Proof We verify the conditions (A1)–(A6).

(A1)—K ′ = F (K) ∪ J is clearly a small subset.

(A2)—J ′ ⊂ K ′ by definition.

(A3)—since F is a left Quillen functor, it preserves cofibrations. But

by hypothesis K ⊂ cof(I), so F (K) ⊂ cof(I ′). Also J ′ ⊂ cof (I ′) so

K ′ ⊂ cof(I ′). The domains of arrows in F (K) are cofibrant again be-

cause F preserves cofibrant objects.

(A4)—if X ∈ R′ and X ∼= Y in ho(M ), choose a cofibrant (resp. fi-

brant) replacement by a weak equivalence X ← X1 (resp. Y → Y1) in

N . The isomorphism in the homotopy category lifts to a diagram of the

form X1 ← Z → Y1 where both maps are weak equivalences, and the

first is a fibration. Hence Z is fibrant, and E(X1) ← E(Z) → E(Y1)

is a diagram of weak equivalences in M . The condition X ∈ R′ means

E(X1) ∈ R. Condition (A4) for R says that E(Z), E(Y1) ∈ R hence

11.8 Transfer along a left Quillen functor 225

Y ∈ R′.

(A5)—Suppose Y ∈ inj(K ′). In particular Y ∈ inj(J ′) so Y is fibrant.

By the adjunction between F and E, the lifting property for Y with re-

spect to F (K) is equivalent to the lifting property for E(Y ) with respect

to K. Hence E(Y ) ∈ inj(K) so E(Y ) ∈ R. Here Y is its own fibrant

replacement so this shows that Y ∈ R′.

We now get to the important part of the argument, which is the ver-

ification of (A6). Suppose X ∈ R′ and X is fibrant (i.e. J ′-injective),

and that X → Y is a pushout by an element of K ′. If it is a pushout

by an element of J ′ then it is already a weak equivalence so we can take

Y = Z (the identity is by definition in cell(K ′)).

Suppose it is a pushout by an element F (g) : F (A) → F (B) where g ∈

K. That is to say we have a map i′ : F (A) → X and Y = X∪F (A)F (B).

By adjunction this gives a map i : A → E(X). Since X is its own

fibrant replacement, by definition of R′ we have E(X) ∈ R. Also E is

a right Quillen functor so it preserves fibrant objects: E(X) is fibrant.

Apply (A6) to the pushout E(X) → E(X) ∪A B in M . That gives a

map h : E(X) ∪A B → Z in cell(K) such that E(X) → Z is a weak

equivalence. Note that both g and h are cofibrations, so F (g) and F (h)

are cofibrations, and F (h) : F (E(X) ∪A B) → F (Z) is in cell(K).

The pushout E(X) → E(X) ∪A B is a cofibration so the composed

map E(X) → Z is a cofibration, hence a trivial cofibration. These are

preserved by F so F (E(X)) → F (Z) is a trivial cofibration.

The adjunction map F (E(X)) → X induces a map F (E(X)) ∪A

B) → X ∪F (A) F (B) and the pushout of F (h) along here gives a map

Y = X ∪F (A) F (B) → (X ∪F (A) F (B)) ∪F (E(X)∪AB) F (Z) =: Z ′

in cell(K). On the other hand,

Z ′ = (X ∪F (A) F (B)) ∪F (E(X)∪AB) F (Z) = X ∪F (E(X)) F (Z),

but that is a pushout along the trivial cofibration F (E(X)) → F (Z), so

the map X → Z ′ is a trivial cofibration. This completes the verification

of (A6).

To show that F is a left Quillen functor, note that the cofibrations

are the same in both cases so we just have to show that F preserves

new trivial cofibrations. Suppose Xf→ Y is a new trivial cofibration. It

fits into a diagram as in Corollary 11.4.4. Applying the functor F takes

cell(K) to cell(K ′) so F (f), which is a cofibration by the original left

Quillen property of F , again fits into a diagram of the same form. Thus

by Corollary 11.4.4, F (f) is a new trivial cofibration.


Applying the last paragraph of Theorem 11.7.1, which in turn comes

from Proposition 9.9.8, gives the following remark. It is the abstract

version appropriate to the present stage of our construction, of Bergner’s

result characterizing fibrant Segal categories [36].

Remark 11.8.2 In the situation of Theorem 11.8.1, an object U ∈

N is fibrant for the new model structure corresponding to (R′,K ′), if

and only if it is fibrant in the original structure of N and E(U) is K-

injective. If U is a new fibrant object of N then a morphism Wp→ U is

a new fibration if and only if it is an old fibration and E(p) is in inj(K).

Suppose now that we have a set Q and a collection of tractable left

proper model categories Mq for q ∈ Q, with a collection of Quillen func-

tors Fq : Mq ←→ N : Eq to a fixed tractable left proper N . Suppose

(Rq,Kq) are direct localizing systems for the Mq. Let (I ′, J ′) be gen-

erators for N , and define (R′,K ′) as follows. First, R′ ⊂ N is the

full subcategory of objects Y ∈ N such that for a fibrant replacement

Y → Y ′, we have EE =q (Y′) ∈ Rq for all q ∈ Q. Then let

K ′ := J ′ ∪⋃

q∈Q

Fq(Kq).

Theorem 11.8.3 In this situation, (R′,K ′) is a direct localizing sys-

tem for N . An object U ∈ N (resp. a morphism Wp→ U to a fibrant

object) is fibrant for the resulting new model structure (resp. a new fi-

bration), if and only if it is fibrant in the old structure and if each Eq(U)

(resp. Eq(p)) is fibrant in the direct left Bousfield localization correspond-

ing to (Rq,Kq).

Proof The same as above. For (A6) note that we have to treat pushout

by a single map in K ′, which is either in J ′ or of the form Fq(g) for

g ∈ Kq. Use the argument of the previous proof for the functors Fq and

Eq.

The characterization of fibrant objects and fibrations to fibrant ob-

jects, comes from the corresponding part of Theorem 11.7.1 going back

to Proposition 9.9.8.

PART III

GENERATORS AND RELATIONS

12

Precategories

This chapter introduces the main object of study, the notion of M -

precategory. The terminology “precategory” has been used in several

different ways, notably by Janelidze [122]. The idea of the word is to in-

voke a structure coming prior to the full structure of a category. For us, a

categorical structure means a category weakly enriched over M following

Segal’s method. Then a “precategory” will be a kind of simplicial object

without imposing the Segal conditions. The passage from a precategory

to a weakly enriched category consists of enforcing the Segal conditions

using the small object argument. The basic philosophy behind this con-

struction is that the precategory contains the necessary information for

defining the category, by generators and relations. The small object ar-

gument then corresponds to the calculus whereby the generators and

relations determine a category, this operation being generically denoted

Seg. Splitting up things this way is motivated by the fact that simplicial

objects satisfying the Segal condition are not in any obvious way closed

under colimits. When we take colimits we get to arbitrary simplicial ob-

jects or precategories, which then have to generate a category by the

Seg operation.

The calculus of generators and relations is the main subject of several

upcoming chapters. In the present chapter, intended as a reference, we

introduce the definition of precategory appropriate to our situation, and

indicate the construction of some important examples which will be used

later. For the purposes of the present chapter we don’t need to be too

specific about the hypotheses on M ; it will generally be supposed to be a

tractable left proper cartesian model category, but this will be discussed

in detail in Chapter 14.


230 Precategories

12.1 Enriched precategories with a fixed set of

objects

Suppose X is a set. Following Lurie [155], define the category ∆X to

have objects which are sequences of xi ∈ X denoted by (x0, . . . , xn) for

any n ∈ ∆, and morphisms

(x0, . . . , xn)φ→ (y0, . . . , ym)

for any φ : n → m in ∆, whenever yϕ(i) = xi for i = 0, . . . , n. Write ∆oX

for the opposite category.

If f : X → Y is a map of sets, we obtain a functor ∆f : ∆X → ∆Y

defined by ∆f (x0, . . . , xn) := (f(x0), . . . , f(xn)) and the corresponding

functor on opposite categories denoted ∆of : ∆o

X → ∆oY .

The basic objects of study will be functors F : ∆oX →M . Such a

functor specifies for each sequence of elements x0, . . . , xn ∈ X , an object

F(x0, . . . , xn) and for each arrow φ : n → m and sequence y0, . . . , ym a

morphism

F(y0, . . . , ym) → F(yφ(0), . . . , yφ(n))

compatible with compositions and identities on the level of φ. Recall

that the category of such functors is denoted Func(∆oX ,M ).

For reasons which will become clear with the counterexample of Sec-

tion 19.3.1 in the later chapter on products, we want to impose a unitality

condition, saying that F(y0) = ∗ for single-object sequences. The uni-

tality condition corresponds, in a certain sense, to requiring strict units

even though the composition of morphisms is not yet specified.

Definition 12.1.1 An M -precategory overX is a functor F : ∆oX →M

such that F(x0) ∼= ∗ is the coinitial object of M , for any x0 ∈ X. Let

PC(X ;M ) denote the category of M -precategories over X with mor-

phisms which are natural transformations of diagrams.

We occasionally use various terminologies such as weakly M -enriched

precategory with object set X , or some subset of those words, for elements

of PC(X ;M ).

In Section 13.4 of the next chapter we shall consider more generally a

category of diagrams with unitality condition denoted Func(Φ/Φ0,M )

whenever Φ is a small category and Φ0 a subset of objects or equiva-

lently a full subcategory. In the present case Φ = ∆oX and the subset Φ0

consists of all sequences of length zero (x0). Note that Φ0 is isomorphic

to the discrete category corresponding to the set X , so in the notation

12.2 The Segal conditions 231

of Section 13.4,

PC(X ;M ) := Func(∆oX/X,M ).

It would of course be interesting to investigate further what would

happen if we considered all objects in Func(∆oX ,M ), with a weak uni-

tality condition imposed by the Segal condition for n = 0, saying that

F(x0) should be contractible. For the present purposes this doesn’t lead

directly to a cartesian model category, indeed the degeneracies of the

single points F(x0) = ∗ play an important role in assuring the compat-

ibility of weak equivalences with products. It is likely that this problem

could be worked around, and that the resulting theory would be closely

related to Kock’s weak unit condition [140] as well as to Paoli’s special

Catn-groups.

For the present exposition it will be convenient to proceed as directly

as possible without considering these other possibilities. When necessary,

we refer to the objects of Func(∆oX ,M ) as “non-unital precategories”.

12.2 The Segal conditions

Given an M -precategory A over a set X , we can look at the Segal maps.

If (x0, . . . , xn) is a sequence in X , the principal edges of the n-simplex

are maps in ∆X

(xi, xi+1) → (x0, . . . , xn)

and these give maps

A(x0, . . . , xn) → A(xi, xi+1).

Put these together to get the Segal map at (x0, . . . , xn)

A(x0, . . . , xn) → A(x0, x1)× · · · × A(xn−1, xn).

Note that for n = 0 the Segal map at (x0) is

A(X0) → ∗ .

Thus the unitality condition says that the n = 0 Segal maps are isomor-

phisms.

Say that an M -precategory satisfies the Segal condition if the Segal

maps are weak equivalences, in other words they are contained in the

subcategory of weak equivalences W ⊂ M . A precategory satisfying

this condition will be called an weakly M -enriched category or just weak

232 Precategories

M -category. An M -precategory is said to be a strict M -category if the

Segal maps are isomorphisms.

As was amply pointed out in Part I, the Segal condition at n = 2

serves to define a weak composition operation in the following sense.

For any three objects x0, x1, x2 ∈ X we have a diagram

A(x0, x1, x2) → A(x0, x1)×A(x1, x2)

A(x0, x2)

↓

where the horizontal arrow is the Segal map; if it is a weak equivalence

then the vertical arrow projects to a map

A(x0, x1)×A(x1, x2) → A(x0, x2)

in ho(M ). The Segal conditions for higher n serve to fix the higher

homotopy coherences necessary starting with associativity at n = 3. It

is necessary to include all of the higher coherences in order to obtain a

theory compatible with products, see Chapter 19 below.

12.3 Varying the set of objects

Up until now we have discussed the category of M -precategories on a

fixed set of objects. This way of splitting off a first part of the argument

was originally suggested by Barwick [14], and that idea will be continued

in Section 14.1 where we consider model structures on PC(X,M ).

The next step is to investigate what happens under maps between sets

of objects, and to define a category PC(M ) of M -precategories without

specified set of objects.

If f : X → Y is a map of sets, we can pull back a structure of

M -precategory on Y , to a structure of M -precategory on X , just by

composing a diagram A : ∆oY →M with the functor ∆o

f :

(∆of )

∗ : Func(∆oY ,M ) → Func(∆o

X ,M ),

(∆of )

∗(A)(x0, . . . , xn) = A(f(x0), . . . , f(xn)).

This clearly preserves the unitality condition, so it restricts to a functor

which, if no confusion arises, will be denoted just by

f∗ : PC(Y ;M ) → PC(X ;M ).

12.3 Varying the set of objects 233

We also get a left adjoint to f∗ denoted f! : PC(X ;M ) → PC(Y ;M ).

If f : X → Y is an inclusion, then we can write Y = f(X) ⊔ Z where

Z is the complement of the image. In this case, f!(A)(y0, . . . , yn) =

A(x0, . . . , xn) if all yi ∈ f(X) and xi is the preimage of yi. If y0 = . . . =

yn = z ∈ Z then f!(A)(y0, . . . , yn) = ∗, and f!(A)(y0, . . . , yn) = ∅ in all

other cases. Thus, f!(A) is the precategory A, extended by adding on

the discrete set Z considered as a discrete M -enriched category.

Lemma 12.3.1 Suppose f : X → Y is a map of sets. If A is an

M -precategory on object set Y , which satisfies the Segal condition, then

f∗(A) satisfies the Segal condition as an M -precategory on X.

Assume at least condition (DCL) which is part of the cartesian con-

dition 10.0.9 on M . If f is injective and if B is an M -precategory on

X satisfying the Segal condition, then f!(B) satisfies the Segal condition

as an M -precategory on Y .

Proof The Segal maps for f∗(A) are some among the Segal maps for

A, so the Segal conditions for A imply the same for f∗(A). Suppose

f is injective and B ∈ PC(X,M ) satisfies the Segal conditions. Use

the notations Y = f(X) ⊔ Z of the paragraph before the lemma. Given

a sequence (y0, . . . , yn) of objects yi ∈ Y , if yi = f(xi) for all i then

the Segal map for f!(B) at (y0, . . . , yn) is the same as that for A at

(x0, . . . , xn). If y0 = . . . = yn = z ∈ Z then the Segal map for f!(B) at

(y0, . . . , yn) is the identity of ∗. If (y0, . . . , yn) is a sequence which is not

constant and which contains at least one element of Z, then one of the

adjacent pairs (yi−1, yi) has to be nonconstant and contain an element

of Z. Using condition (DCL) via Lemma 10.0.10, the direct product of

anything with ∅ is again ∅, so in these cases the Segal maps are the

identity of ∅. In all cases, the Segal maps remain weak equivalences.

Lemma 12.3.2 Suppose f : X → Y is a map of sets. The pullback

f∗ preserves levelwise weak equivalences, that is it takes levelwise weak

equivalences in PC(Y,M ) to levelwise weak equivalences in PC(X,M ).

Similarly, f∗ preserves levelwise cofibrations and trivial cofibrations.

Proof The pullback f∗ of diagrams will preserve any levelwise proper-

ties.

234 Precategories

12.4 The category of precategories

We would like to define a notion of M -precategory on an unspecified or

variable set of objects, just as for usual categories. An M -precategory

A consists of a set denoted Ob(A), and an M -precategory over Ob(A)

denoted

(x0, . . . , xn) 7→ A(x0, . . . , xn) ∈M .

A morphism f : A → B between two M -precategories consists of a

map of sets Ob(f) : Ob(A) → Ob(B), and for any x0, . . . , xn ∈ Ob(A)

a morphism

f(x0, . . . , xn) : A(x0, . . . , xn) → B(Ob(f)(x0), . . . ,Ob(f)(xn));

these are required to satisfy naturality with respect to maps φ : n → m

in ∆. Henceforth, if no confusion is possible, we denote by f(xi) :=

Ob(f)(xi) the action on objects, and use f also to denote f(x0, . . . , xn).

In terms of the notation given in Section 12.3, we can think of f as a

morphism of M -precategories over object set Ob(A), denoted

Morf : A → Ob(f)∗(B).

Composition of morphisms is defined in the obvious way.

Let PC(M ) denote the category of M -precategories thusly defined.

Taking the “underlying set of objects” is a functor

Ob : PC(M ) → Set.

This is a fibered category as we now explain. Given a map of sets g :

X → Y this induces a functor ∆g : ∆X → ∆Y hence an adjoint pair

∆g,! : Func(∆oX ;M ) ←→ Func(∆o

Y ;M ) : ∆∗g.

We have ∆∗g(F)(x0, . . . , xn) = F(g(x0), . . . , g(xn)). The adjoint pair on

the categories of M -precategories is

PCg,! : PC(X ;M ) ←→ PC(Y ;M ) : ∆∗g.

with PCg,! = UY,! ∆g,!,u.

Consider the fibered categoryF → Setwhose fiber overX isPC(X ;M )

and whose pullback maps are the ∆∗g. An object of F is by definition

a pair (X,A) where X ∈ Set and A ∈ PC(X ;M ). A morphism from

(X,A) to (Y,B) is a pair (g, h) where g : X → Y is a morphism in Set

and h : A → g∗(B) = ∆∗g is a morphism in PC(Y ;M ). By inspection,

this is the same structure as defined above, that is F = PC(M ), with

12.5 Basic examples 235

functor Ob being the projection to Set. A morphism f = (Ob(f), f) is

cartesian if and only if Morf induces isomorphisms

f(x0, . . . , xn) : A(x0, . . . , xn)∼=→ B(Ob(f)(x0), . . . ,Ob(f)(xn)).

12.5 Basic examples

In this section we indicate many of the basic examples of precategories

which will be important at various places in the subsequent chapters.

Rather than give a full definition and discussion here, we refer to the

appropriate places when necessary.

Suppose X is a set. The discrete precategory disc(X) is defined to

have Ob(disc(X)) := X and

disc(X)(x0, . . . , xn) :=

∗ if x0 = · · · = xn∅ otherwise

Here ∅ and ∗ denote the initial and coinitial objects of M respectively.

The functor X 7→ disc(X) provides a functor Set → PC(M ), fully

faithful and left adjoint to Ob : PC(M ) → Set.

The codiscrete precategory codsc(X) is defined to have Ob(codsc(X)) :=

X and

codsc(X)(x0, . . . , xn) := ∗

for all sequences x·. The functor codsc : Set → PC(M ) is fully faithful

and right adjoint to Ob.

Suppose k ∈ N. Denote by

[k] := υ0, . . . , υk, υ0 < υ1 < · · · < υk

the standard linearly ordered set with k + 1 elements. The [k] are the

objects of ∆. It will be important to consider various different precate-

gories whose underlying set of objects is [k]. These examples will all be

ordered precategories in the sense that if (y0, . . . , yp) is any sequence of

elements of the set [k] with yj = υij , and if the sequence y· is strictly

decreasing at any place (i.e. if it is not an increasing sequence), then

A(y0, . . . , yp) = ∅.

For B ∈M we have the precategory h([k], B) which has the following

concrete description (see Proposition 15.2.1): suppose (y0, . . . , yp) is any

sequence of elements of the set [k] with yj = υij . Then:

236 Precategories

—if (y0, . . . , yp) is increasing but not constant i.e. ij−1 ≤ ij but i0 < ipthen

h([k];B)(y0, . . . , yp) = B;

—if (y0, . . . , yp) is constant i.e. i0 = i1 = . . . = ip then

h([k];B)(y0, . . . , yp) = ∗;

and otherwise, that is if there exists 1 ≤ j ≤ p such that ij−1 > ij then

h([k];B)(y0, . . . , yp) = ∅.

For any precategory A ∈ PC(M ), a map h([k];B) → A is the same

thing as a collection of elements x0, . . . , xk ∈ Ob(A) together with a

map B → A(x0, . . . , xk) in M , so we can think of h([k], B) as being a

“representable” object in a certain sense.

It has as “boundary” the precategory h(∂[k], B), defined using the

skeleton construction in Chapter 15, with the following description (see

Lemma 15.2.3):

—if (y0, . . . , yp) is increasing but not constant i.e. ij−1 ≤ ij but i0 < ip,

and if there is any 0 ≤ m ≤ k such that ij 6= m for all 0 ≤ j ≤ k, then

h(∂[k];B)(y0, . . . , yp) = B;


h(∂[k];B)(y0, . . . , yp) = ∗;

and otherwise, that is if either there exists 1 ≤ j ≤ p such that ij−1 > ijor else if the map j 7→ yj is a surjection from 0, . . . , p to [k], then

h(∂[k];B)(y0, . . . , yp) = ∅.

More generally, the pushouts

h([k], ∂[k];Af→ B) := h([k];A) ∪h(∂[k];A) ∂h(∂[k];B)

are also useful, being the generators of the Reedy cofibrations inPC(M ).

In Section 16.1 and later in Chapter 18 we consider precategories

Υ(B1, . . . , Bk) with the same set of objects [k], depending onB1, . . . , Bk ∈

M . The basic idea is to put Bi in as space of morphisms from υi−1 to

υi. Thus, the main part of the structure of precategory is given by

Υ(B1, . . . , Bk)(υi−1, υi) := Bi.


This is extended whenever there is a constant string of points on either

side:

Υ(B1, . . . , Bk)(υi−1, . . . , υi−1, υi, . . . , υi) := Bi.

The unitality condition on the diagram ∆υ0,...,υk →M means that

for 0 ≤ i ≤ k we have

Υ(B1, . . . , Bk)(υi, . . . , υi) := ∗.

In all other cases,

Υ(B1, . . . , Bk)(x0, . . . , xn) := ∅.

If A ∈ PC(M ), a map Υ(B1, . . . , Bk) → A is the same thing as a collec-

tion of objects x0, . . . , xk ∈ Ob(A) together with maps Bi → A(xi−1, xi)

in M .

The categorification of Υ(B1, . . . , Bk) can be described explicitly; it is

denoted by Υ(B1, . . . , Bk) in Section 18.1. For any sequence υi0 , . . . , υinwith i0 ≤ . . . ≤ in, we put

Υk(B1, . . . , Bk)(υi0 , . . . , υin) := Bi0+1 ×Bi0+2 × · · · ×Bin−1 ×Bin .

(12.5.1)

For any other sequence, that is to say any sequence which is not increas-

ing, the value is ∅. The value on a constant sequence is ∗.

As a rough approximation, the calculus of generators and relations

can be understood as being the small object argument applied using

the inclusions Υ(B1, . . . , Bk) → Υ(B1, . . . , Bk). Starting with a precat-

egory A, for every map Υ(B1, . . . , Bk) → A, take the pushout with

Υ(B1, . . . , Bk). Keep doing this and eventually we get to a precategory

which satisfies the Segal conditions.

12.6 Limits, colimits and local presentability

It will be useful to have explicit descriptions of limits and colimits in

PC(M ). We can then show local presentability.

Suppose Aii∈α is a diagram of objects Ai ∈ PC(M ), that is a

functor α → PC(M ). Let Xi := Ob(Ai) denote the object sets so

we can consider Ai ∈ PC(Xi;M ). For any f : i → j in α denote by

φf : Xi → Xj the transition map on object sets, then ρf : Ai → φ∗fAjthe transition maps on the level of precategories.

238 Precategories

Start by constructing the limit in PC(M ). The object set of the limit

will be

X := limi∈αXi.

We have maps pi : X → Xi, hence p∗i (Ai) ∈ PC(X ;M ). These are

provided with transition maps, indeed for f : i → j in α, φfpi = pj so

p∗i (φ∗fAj) = p∗jAj and p∗i (ρf ) : p∗i (Ai) → p∗jAj provide the transition

maps for the system of p∗i (Ai) considered as a diagram α → PC(X ;M ).

Put

A := limPC(X;M )i∈α p∗i (Ai) ∈ PC(X ;M ).

We claim that this is the limit of the diagram Aii∈α in PC(M ).

Suppose B ∈ PC(M ) with Ob(B) = Y and suppose given a sys-

tem of maps B → Ai. These correspond to maps ri : Y → Xi and

ϕi : B → r∗i (Ai). The collection of ri gives a uniquely determined map

r : Y → X , and r∗p∗i (Ai) = r∗i (Ai) so the ϕi correspond to a collec-

tion of maps B → r∗(p∗i (Ai)). This gives a uniquely determined map

ϕ : B → r∗(A) whose composition with the projections of the limit ex-

pression for A, are the ϕi. The pair (r, ϕ) is a map B → A in PC(M )

uniquely solving the universal problem to show that (X,A) is the limit

of the Ai.

The limit A can be described explicitly as an element of PC(X ;M ),

by the discussion preceding Lemma 14.4.2: for any x0, . . . , xn ∈ X ,

A(x0, . . . , xn) = limi∈αAi(pix0, . . . , pixn).

Turn now to the construction of the colimit. The object set will be

Z := colimi∈αXi

with maps qi : Xi → Z. These give qi,∗(Ai) ∈ PC(X ;M ), with transi-

tion maps defined as follows. If f : i → j is an arrow in α then qjφf = qiso qj,!(φf,!(Ai)) = qi,!(Ai). The adjunction between φf,! and φ

∗f means

that the transition map ρf for the system of Ai may be viewed as a map

denoted ρf : φf,!(Ai) → Aj , in particular we get

qj,!(ρf ) : qi,!(Ai) = qj,!(φf,!(Ai)) → qj,!(Aj).

These provide transition maps for the diagram qi,!(Ai)i∈α with values

in PC(Z;M ) and set

C := colimPC(Z;M )i∈α qi,!(Ai) ∈ PC(Z;M ).

The object (Z, C) is the colimit of the diagram of Ai, for the same formal

reason as in the previous discussion of the limit.


The colimit to define C is taken in the category PC(Z;M ) which

presupposes in general applying the operation U! if α is disconnected.

However, when the definition is unwound explicitly this phenomenon

disappears, being absorbed in the calculation of the value of C on a

sequence of points (z0, . . . , zn) via the introduction of a new category

α/(z0, . . . , zn).

Suppose (z0, . . . , zn) is a sequence of elements of Z. Let α/(z0, . . . , zn)

denote the set of pairs (i, (x0, . . . , xn)) where i ∈ α and (x0, . . . , xn)

is a sequence of points in Xi such that qi(xk) = zk for k = 0, . . . , n.

The association (i, (x0, . . . , xn)) 7→ Ai(x0, . . . , xn) is a diagram from

α/(z0, . . . , zn) to M .

Lemma 12.6.1 In the above situation, the value of the colimiting ob-

ject C on the sequence (z0, . . . , zn) is calculated as a colimit of the dia-

gram α/(z0, . . . , zn) →M :

C(z0, . . . , zn) = colim(i,(x0,...,xn))∈α/(z0,...,zn)Ai(x0, . . . , xn).

Proof Let qwui,! : Func(∆oXi,M ) → Func(∆o

Z ;M ) denote the push-

forward in the world of non-unital precategories. It is the pushforward

for diagrams valued in M along the functor

∆oXi→ ∆o

Z .

If (z0, . . . , zn) is a sequence of points in Z, then qwui,! (Ai)(z0, . . . , zn) is the

colimit of the Ai(x0, . . . , xk) over the category of pairs (u, (x0, . . . , xk))

where u : (qix0, . . . , qixk) → (z0, . . . , zn) is a map in ∆oZ or equivalently

(z0, . . . , zn) → (qix0, . . . , qixk) is a map in ∆Z . Such a map factors

through a unique map (z0, . . . , zn) → (qixi0 , . . . , qixin) where (i0, . . . , in)

is a multiindex representing a map [n] → [k] in ∆. Hence the category of

pairs in question is a disjoint union of categories having coinitial objects.

This yields the expression of qwui,! (Ai)(z0, . . . , zn) as the disjoint sum of

Ai(x0, . . . , xn) over all sequences (x0, . . . , xn) of objects in Xi such that

qi(xj) = zj. Putting these together over all i ∈ α gives

colimi∈α(qwui,! (Ai)(z0, . . . , zn)) = colim(i,(x0,...,xn))∈α/(z0,...,zn)Ai(x0, . . . , xn).

Now U! is a left adjoint so it preserves colimits, in particular

C = colimPC(Z;M )i∈α qi,!(Ai) = colimi∈αU!(q

wui,! (Ai))

= U!

(colimi∈αq

wui,! (Ai)

)= U!(C

′)

240 Precategories

where C′ ∈ Func(∆oZ ;M ) is the object defined by

C′(z0, . . . , zn) := colim(i,(x0,...,xn))∈α/(z0,...,zn)Ai(x0, . . . , xn).

To finish the proof note that C′ is already in PC(Z;M ). Indeed for a

single element z0 ∈ Z the category α/(z0) of pairs (i, x0) with qi(x0) = z0is connected. This is factoid about colimits of sets. Since Ai is unital we

have Ai(x0) = ∗, thus

C′(z0) = colim(i,x0)∈α/(z0)∗ = ∗,

in other words C′ is unital. Therefore C = U!(C′) = C′ which is the

statement of the lemma. An alternative proof would be to note that C′

satisfies the required universal property for defining a colimit.

Corollary 12.6.2 If α is a connected category and A· : α → PC(X ;M )

is a diagram of M -precategories with a common object set X, then the

natural map

colimPC(M )i∈α Ai → colim

PC(X;M )i∈α Ai

is an isomorphism.

Proof The explicit descriptions of both sides are the same.

It is worthwhile to discuss explicitly some special cases. For example

Ob(A×B) = Ob(A)×Ob(B) and for any sequence ((x0, y0), . . . , (xn, yn))

of elements of Ob(A)×Ob(B) we have

A× B((x0, y0), . . . , (xn, yn)) = A(x0, . . . , xn)× B(y0, . . . , yn).

This extends to fiber products: if A → C and B → C are maps, then

Ob(A×C B) = Ob(A) ×Ob(C) Ob(B)

and again for any sequence ((x0, y0), . . . , (xn, yn)) of elements in the fiber

product of object sets we have

A×C B((x0, y0), . . . , (xn, yn)) = A(x0, . . . , xn)×C(z0,...,zn) B(y0, . . . , yn)

where the zi are the common images of xi and yi in Ob(C).

For colimits, the disjoint sum A ⊔ B has object set Ob(A) ⊔ Ob(B),

and for a sequence of elements (z0, . . . , zn) in here we have

(A ⊔ B)(z0, . . . , zn) =

A(z0, . . . , zn) if all zi ∈ Ob(A)

B(z0, . . . , zn) if all zi ∈ Ob(B)

∅ otherwise

.

Look at the coproduct or pushout of two maps u : C → A and v : C → B,


supposing that one of the maps say v is injective on the set of objects.

This will usually be the case in our applications because we usually look

at pushouts along cofibrations. The coproduct A∪C B has object set

Ob(A) ∪Ob(C) Ob(B) = Ob(A) ⊔ (Ob(B)−Ob(C)).

The category α indexing the colimit has three objects denoted a, b, c with

arrows a ← c → b. Given a sequence (z0, . . . , zn) of elements in here,

the category α/(z0, . . . , zn) has objects of three kinds denoted a, b and

c, and objects of a given kind correspond to sequences in Ob(A), Ob(B)

or Ob(C) respectively mapping to the given z·. Our general formula of

Lemma 12.6.1 expresses (A ∪C B)(z0, . . . , zn) as the pushout of disjoint

sums∐

x· 7→z·

A(x0, . . . , xn) ∪∐

w· 7→z·C(w0,...,wn)

∐

y· 7→z·

B(y0, . . . , yn).

Another important case is that of filtered colimits. Suppose α is a

filtered (resp. κ-filtered) category and Aii∈α is a diagram in PC(M ).

PutXi := Ob(Ai) and Z := colimi∈αXi. Then for any sequence (z0, . . . , zn)

of elements of Z, the category α/(z0, . . . , zn) is again filtered (resp. κ-

filtered), so

A(z0, . . . , zn) = colim(i,(x0...,xn)∈α/(z0,...,zn)A(x0 . . . , xn)

is a filtered (resp. κ-filtered) colimit in M .

Lemma 12.6.3 Suppose M is locally presentable. An object A ∈

PC(M ) is κ-presentable if and only if X := Ob(A) is a set of car-

dinality < κ, and each A(x0, . . . , xp) is a κ-presentable object of M .

Proof Suppose X := Ob(A) is a set of cardinality < κ, and each

A(x0, . . . , xp) is a κ-presentable object of M . Suppose Bii∈β is a dia-

gram in PC(M ) indexed by a κ-filtered category β, and suppose given

a map

A → B := colimi∈βBi.

In particular we get a map Ob(A) → Ob(B) = colimi∈βOb(Bi) and

the condition |Ob(A)| < κ implies that this map factors through a

map Ob(A) → Ob(Bj) for some j ∈ β. Given a sequence of objects

(x0, . . . , xn) ∈ Ob(A), let (y0, . . . , yn) denote the corresponding sequence

of objects in Ob(Bj) and (z0, . . . , zn) the sequence in Ob(B). Let j\α

denote the category of objects under j in α. Given j → i the image

242 Precategories

of (y0, . . . , yn) is a sequence denoted (yi0, . . . , yin) in Ob(Bi) mapping to

(z0, . . . , zn). This gives a functor

j\α → α/(z0, . . . , zn),

and the κ-filtered property of α implies that this functor is cofinal. Hence

by Lemma 12.6.1 and the invariance of colimits under cofinal functors,

B(z0, . . . , zn) = colim(j→i)∈j\αBi(yi0, . . . , y

in).

The category j\α is κ-filtered, so the map

A(x0, . . . , xn) → B(z0, . . . , zn)

factors through one of the Bi(yi0, . . . , y

in) for j → i in α. The cardi-

nality of the set of possible sequences (x0, . . . , xn) is < κ, so the κ-

filtered property says that we can choose a single i. Then the stan-

dard kind of argument shows that by going further along, the maps

A(x0, . . . , xn) → Bi(yi0, . . . , yin) can be assumed to all fit together into

a natural transformation in terms of (x0, . . . , xn) ∈ ∆oX . Thus we get

a factorization of our map A → B through some A → Bi. This shows

that A is κ-presentable.

Suppose on the other hand that A is κ-presentable. We note first

of all that |X | < κ. Indeed, if |X | ≥ κ then we could consider a κ-

filtered system of subsets Zi ⊂ X with |Zi| < κ, but colimZi = X .

Let codsc(Zi) and codsc(X) denote the codiscrete precategories on

these object sets, that is the precatgories whose value is ∗ on any se-

quence of objects. Then colimicodsc(Zi) = codsc(X) in PC(M ) (see

Lemma 8.1.8), but the identity map on underlying object sets gives a

mapA → codsc(X) not factoring through any codsc(Zi), contradicting

the assumed κ-presentability of A. Hence we may assume that |X | < κ.

We claim that A is κ-presentable when considered as an object in

PC(X ;M ). Indeed, if Bii∈β is a κ-filtered diagram in PC(X ;M )

then since β is connected, any map

A → colimPC(X;M )i∈β Bi

is also a map A → colimPC(M )i∈β Bi by Corollary 12.6.2. By the assumed

κ-presentability of A this would have to factor through one of the Bi,

necessarily as a map inducing the identity on underlying object sets

X . This shows that A is κ-presentable when considered as an object

in PC(X ;M ). Now Corollary 14.4.3 tells us that the A(x0, . . . , xp) are

κ-presentable objects of M .


Proposition 12.6.4 Suppose M is locally presentable. Then the cat-

egory of M -precategories PC(M ) is also locally presentable.

Proof Let κ be a regular cardinal such that M is locally κ-presentable.

Note that, for any setX the categoryPC(X,M ) is locally κ-presentable

by its adjunction with the diagram category Func(∆oX ,M ) which in

turn is locally κ-presentable by Lemma 8.1.3.

Existence of arbitrary colimits in PC(M ) was shown at the start

of the section. It is clear from the description of κ-presentable objects

in Lemma 12.6.3 that the isomorphism classes of κ-presentable objects

of PC(M ) form a set. Furthermore, any object (Z,A) is κ-accessible.

Indeed, consider the category of triples (X,B, u) whereX ⊂ Z is a subset

of cardinality |X | < κ, B ∈ PC(X,M ) is a κ-presentable object, and u :

B → A|X is a morphism to the pullback of A along the inclusion X ⊂ Z.

Using the local κ-presentability of each PC(X,M ) and the expression

of Z as a κ-filtered union of the subsets X , the category of triples is

κ-filtered and the colimit of the tautological functor is (Z,A).

12.7 Interpretations as presheaf categories

With some additional hypotheses on M , the trick of introducing the

categories ∆X becomes unnecessary. This corresponds more closely with

some previous references [206] [193] [194], and will be useful in estab-

lishing notation for iterated n-precatgories. Starting from this first dis-

cussion we show later on that if M itself is a presheaf category then

PC(M ) is a presheaf category.

This consideration will not usually enter into our argument although

it does provide a convenient change of notation for Chapter 17. Never-

theless, many arguments become considerably simpler in the case of a

presheaf category—as we have already seen for cell complexes in Chap-

ter 8. As we shall now see, the passage from M to PC(M ) preserves the

condition of being a presheaf category, and many important initial cases

such as the model category of simplicial sets K satisfy this condition.

So it should be helpful throughout the book to be able to think of the

case of presheaf categories.

The first part of our discussion follows Pelissier [171].

Suppose M is a locally presentable category. Define a functor disc :

244 Precategories

Set →M by

disc(U) :=∐

u∈U

∗.

If necessary we shall denote by ∗u the term corresponding to u ∈ U in

the coproduct. The discrete object functor has a right adjoint disc∗ :

M → Set defined by disc∗(X) := HomM (∗, X), with

HomM (disc(U), X) = HomSet(U,disc∗(X)).

In particular, disc commutes with colimits.

Given an expression X =∐u∈U Xu the universal property of the

coproduct applied to the maps Xu → ∗u yields a map X → disc(U).

On the other hand, given a map f : X → disc(U) we can put

f−1(u) := X ×disc(U) ∗u.

We have a natural map∐u∈U f

−1(u) → X compatible with the maps

to disc(U).

Consider the following hypothesis on M , saying that disjoint unions

(i.e. colimits over discrete categories) behave well. This kind of hypoth-

esis was introduced for the same purpose by Pelissier [171, Definition

1.1.4].

Condition 12.7.1 (DISJ)

(a)—If f : X → disc(U) is a map, then the natural map is an isomor-

phism∐u∈U f

−1(u) ∼= X. If X =∐u∈U Xu is a coproduct expression

then Xu = f−1(u) for the corresponding map f : X → disc(U).

(b)—The coinitial object ∗ is indecomposable, that is to say it cannot be

written as a coproduct of two nonempty objects.

(c)—The category M has more than just a single object up to isomor-

phism.

To see why condition (b) is necessary, note for example that if M =

Presh(Φ) is the category of presheaves on a category Φ which has more

than one connected component (for example the discrete category with

two objects) then ∗ is decomposable.

Condition (c) is required to rule out the trivial category M = ∗ which

satisfies all of our other hypotheses.

Lemma 12.7.2 Assume that M is locally presentable, and satisfies

Condition (DCL) of the cartesian condition 10.0.9. Assume that M sat-

isfies Condition (DISJ) 12.7.1 above. Then we have the following further

properties:


(1)—For any object X ∈ M , giving an expression X ∼=∐u∈U Xu is

equivalent to giving a map X → disc(U).

(2)—If Y →∐u∈U Xu is a map from a single object to a coproduct,

then setting Yu := Y ×∐u∈U Xu

Xu the natural map∐u∈U Yu → Y is

an isomorphism.

(3)—The map ∅ → ∗ is not an isomorphism.

(4)—For any set U the adjunction map U → disc∗disc(U) is an iso-

morphism.

(5)—The functor disc is fully faithful, and disc∗ gives an inverse on its

essential image.

(6)—If X ∼=∐u∈U Xu and Y ∼=

∐u∈U Yu are decompositions corre-

sponding to maps X,Y → disc(U) then

X ×disc(U) Y =∐

u∈U

Xu × Yu.

(7)—Coproducts are disjoint: if Xuu∈U is a collection of objects and

u 6= v then

Xu ×∐u∈U Xu

Xv = ∅.

(8)—The functor disc preserves finite limits.

Proof Condition (1) is just a restatement of the first part of (DISJ).

For (2), suppose Y → X =∐u∋U Xu is a map. Compose with the

map X → disc(U) given by (1), to get Y → disc(U). In turn this

corresponds to a decomposition Y =∐u∈U Yu with

Yu = Y ×disc(U) ∗u,

but Xu = X×disc(U)∗u so Yu = Y ×XXu which is the desired statement.

For (3), if ∅ → ∗ were an isomorphism, then we would have ∅×X = X

for all objects X , but in view of Lemma 10.0.10 following from (DCL)

this would imply that all objects are ∅ contradicting the nontriviality

hypothesis (DISJ) (c) on M .

For (4) suppose first we are given a map f : ∗ → disc(U). By (1)

this corresponds to a decomposition ∗ =∐u∈U ∗ ×disc(U) ∗u. By Con-

dion (DISJ)(b), all but one of the summands must be ∅. From (3) it

follows that one of the summands is different, so there is unique one

of the summands which is ∗. We get that our map factors through a

unique ∗u. This shows that the adjunction map U → disc∗disc(U) is

an isomorphism.

For (5), suppose given a map f : disc(U) → disc(V ). By property (4)

246 Precategories

there is a unique map Ug→ V compatible with disc∗(f) by the adjunc-

tion isomorphisms. Uniqueness shows that disc is faithful. Applying disc

to this comptibility diagram and composing with the naturality square

for the other adjunction map gives a square

disc(U)disc(g)

→ disc(V )

disc(U)

↓f→ disc(V )

↓

where the vertical maps are the adjunction compositions

disc(U) → discdisc∗disc(U) → disc(U)

and the same for V . These are the identities so f = disc(g). This shows

that disc is fully faithful, and the disc∗ gives an essential inverse by (4).

In the situation of (6), given Zh→ disc(U) corresponding to Z =∐

u∈U Zu, a map Z → X compatible with h is the same thing as a

collection of maps Zu → Xu by (2). Similarly for a map to Y . It follows

that∐u∈U Xu×Yu satisfies the universal property for the fiber product,

giving (6).

For (7), it suffices to show that ∗u ×disc(U) ∗v = ∅ for v 6= u, in view

of the consequence Lemma 10.0.10 of Condition (DCL) saying that no

nonempty object can map to ∅. But ∗u =∐v∈U (∗u)×disc(U) ∗v and, as

was seen in the proof of (4), condition (DISJ) (b) implies that all but

one of these terms must be ∅. Since there is a diagonal map from ∗ to

the term v = u, the terms for v 6= u must be ∅.

For (8), the functor disc preserves finite direct products: given maps

Z → disc(U) and Z → disc(V ) they correspond to decompositions as

in (1). The maps Zu → disc(V ) correspond to decompositions

Zu =∐

v∈V

Zu,v, Zu,v = Zu ×disc(V ) ∗v.

Putting these together over all u ∈ U we get a decomposition of Z

corresponding to a unique map Z → disc(U × V ). This shows that

disc(U×V ) satisfies the universal property to be the product disc(U)×

disc(V ). A similar argument shows that disc preserves equalizers.

Suppose now that M satisfies condition (DISJ) in addition to being

tractable left proper and cartesian, so the properties of the preceding


lemma apply. Given A ∈ PC(M ) define a functor ∆o →M denoted

[n] 7→ An/ by

An/ :=∐

(x0,...,xn)∈Ob(A)n+1

A(x0, . . . , xn).

The functoriality maps are defined using those of A. This has the prop-

erty that A0/ is a discrete object, indeed the unitality conditions say

that A(x0) = ∗ so there is a natural isomorphism A0/∼= disc(Ob(A)),

and we identify A0/ with the set Ob(A), sometimes using the notation

A0.

The property that A0/ is a discrete object is called the constancy

condition [206], closely related to the globular nature of the theory of

n-categories. Let Func([0] ⊂ ∆o,Set ⊂M ) denote the full subcategory

of Func(∆o,M ) consisting of functors which satisfy this constancy con-

dition.

Suppose on the other hand that n 7→ An/ is a functor ∆o →M which

satisfies the constancy condition. Set X := Ob(A) := disc∗(A0/), then

A0/ = disc(Ob(A)). For any sequence x0, . . . , xn ∈ X = Ob(A), we get

a map

(x0, . . . , xn) : ∗ → A0/ × · · · × A0/

and define

A(x0, . . . , xn) := An/ ×A0/×···×A0/∗

where the map An/ → A0/ × · · · × A0/ is obtained using the n + 1

vertices of the simplex [n]. Notice that

A0/ × · · · × A0/ = disc(Ob(A)× · · · ×Ob(A))

since disc preserves finite products (Lemma 12.7.2 (8)). The simplicial

maps for A·/ provide transition maps to make A(· · · ) into a functor

∆pX →M .

Theorem 12.7.3 Suppose M satisfies Condition (DISJ) 12.7.1. The

above constructions provide essentially inverse functors

PC(M ) ←→ Func([0] ⊂ ∆o,Set ⊂M ).

The full subcategory Func([0] ⊂ ∆o,Set ⊂ M ) ⊂ Func(∆o,M ) is

closed under limits and colimits, and the above essentially inverse func-

tors preserve limits and colimits.

248 Precategories

Proof This follows from Lemma 12.7.2: suppose fixed the set of objects

X , then giving An/ with map to the direct product

An/ → disc(X × · · ·X) = disc(X)× · · · × disc(X)

is the same as giving the pieces of the decomposition

An/ =∐

(x0,...,xn)

A(x0, . . . , xn).

For iteration of our basic construction, the following lemmas show

that the above notational reinterpretation is very reasonable.

Lemma 12.7.4 Suppose M satisfies the hypothesis (DCL) of Defini-

tion 10.0.9. Then the category PC(M ) satisfies condition (DISJ).

Proof The discrete precategories are constructed as follows. The coini-

tial object ∗ ∈ PC(M ) has a single element Ob(∗) = ∗ = x and

∗(x, . . . , x) = ∗ ∈ M . Thus, if U is any set then disc(U) calculated in

PC(M ) has object set disc(U) ∼= U , and disc(U)(x0, . . . , xn) = ∗ if

x0 = · · · = xn, otherwise disc(U)(x0, . . . , xn) = ∅ if the sequence is not

constant.

For part (a), suppose A ∈ PC(M ) and A → disc(U) is a map.

Put X := Ob(A), with X → U which corresponds to a decomposition

X =∐u∈U Xu. Let Au ⊂ A be the pullback of A along Xu → X , that

is it is the “full sub-precategory” with object set Xu ⊂ X . This is indeed

A×disc(U) ∗u. The map∐

u∈U

Au → A

is an isomorphism. This follows from the fact that disc(U)(x0, . . . , xn) =

∅ for any nonconstant sequence, plus the consequence Lemma 10.0.10 of

(DCL). On the other hand given a decomposition in coproduct A =∐u∈U Au we get a map A → disc(U) for which the components Au are

the fibers.

Conditions (b) and (c) are easy.

Lemma 12.7.5 Suppose Ψ is a connected category (i.e. its nerve is a

connected space, or equivalently any two objects are joined by a zig-zag of

arrows). Then the category Presh(Ψ) = Func(Ψo;Set) satisfies condi-

tion (DISJ), with the discrete objects being those equivalent to constant

functors. In particular, the model category Set where all morphisms are


fibrations and cofibrations, and the weak equivalences are isomorphisms,

satisfies (DISJ). And the model category K of simplicial sets satisfies

(DISJ).

Proof If U is a set, the discrete presheaf disc(U) is the constant presheaf

x 7→ U on all x ∈ Ψ. For (a), a map A → disc(U) is the same thing

as a decomposition A =∐u∈U Au, as can be seen levelwise. Condition

(b) follows from the connectedness of Ψ (indeed it is equivalent), and

Condition (c) is easy.

We now turn to the further situation where M is a presheaf category,

say M = Presh(Ψ) = Func(Ψo;Set). In this case

Func(∆oX ;M ) = Func(∆o

X ×Ψo;Set) = Presh(∆X ×Ψ)

is again a presheaf category.

We construct a new category denoted C(Ψ), which looks somewhat

like a “cone” on Φ. It is defined by contracting [0] × Ψ ⊂ ∆ × Ψ to

a single object denoted 0. Thus, the objects of C(Ψ) are of the form

([n], ψ) for n ≥ 1 and ψ ∈ Ψ, or the object 0. The morphisms are as

follows:

—there is a single identity morphism between 0 and itself;

—for n ≥ 1 and ψ ∈ Ψ there is a unique morphisms from any ([n], ψ) to

0;

—the morphisms from 0 to ([n], ψ) are the same as the morphisms

[0] → [n] in ∆ (i.e. there are n+ 1 of them); and

—the morphisms from ([n], ψ) to ([n′], ψ′) are of two kinds: either (a, f)

where a : [n] → [n′] is a morphism in ∆ such that a doesn’t factor

through [0] and f : ψ → ψ′ is a morphism in Ψ; or else (a) where

a : [n] → [n′] is a morphism in ∆ which factors as [n] → [0] → [n′].

Composition of morphisms is defined in an obvious way. Notice that the

composition of anything with a morphism which factors through [0] will

again factor through [0], which allows us to define compositions of the

form (a) (a′, f ′) or (a, f) (a′) in the last case. The division of the

morphisms from ([n], ψ) to ([n′], ψ′) into two cases is necessary in order

to define composition.

Proposition 12.7.6 Suppose M = Presh(Ψ) is a presheaf category.

There is a natural isomorphism between Presh(C(Ψ)) and the category

of unital M -precategories PC(M ). Thus, PC(M ) is again a presheaf

category.

Proof Suppose F : C(Ψ)o → Set is a presheaf. Let X := F(0). For

250 Precategories

n ≥ 1 and any (x0, . . . , xn) ∈ ∆X , let A(x0, . . . , xn) be the presheaf on Ψ

which assigns to ψ ∈ Ψ the subset of elements of F([n], ψ) which project

to (x0, . . . , xn) under the n+ 1 projection maps F([n], ψ) → F(ι) = X

corresponding to the n + 1 maps 0 → ([n], ψ). At n = 0, set A(x0) :=

x0. The pair (X,A) is an element of PC(M ). Conversely, given any

(X,A) ∈ PC(M ), define a presheaf F : C(Ψ)o → Set by setting

F(0) := X and

F([n], ψ) :=∐

(x0,...,xn)∈Xn+1

A(x0, . . . , xn)(ψ).

These constructions are inverses.

Let cΨ : ∆×Ψ → C(Ψ) denote the projection. If Ψ is connected, then

the category of presheaves Presh(C(Ψ)) may be identified, via c∗Ψ, with

the full subcategory of Presh(∆ × Ψ) consisting of presheaves A which

satisfy Tamsamani’s constancy condition that A(0, ψ) is a constant set

independent of ψ ∈ Ψ.

The construction Ψ 7→ C(Ψ) was the iterative step in the construction

of the sequence of categories denoted Θn in [193]. In that notation, Θ0 =

∗ and Θn+1 = C(Θn). However, the notation Θn was subsequently used

by Joyal [127] to denote a related but different sequence of categories;

in order to avoid confusion we will use the cone notation C.

For the theory of non-unital M -precategories, one can also construct

a category denoted C+(Ψ) with the property that Presh(C+(Ψ)) is

the category of non-unital M -precategories. This is in fact even more

straightforward than the construction of C for the unital theory. The fol-

lowing discussion is optional, but serves to put the previous discussion

of C into a better perspective.

Recall that PC(M ) was defined as a fibered category over Set whose

fiber over a set X was PC(X,M ). In the same way, the category of

non-unital M -precategories is the fibered category over Set whose fiber

over X is Func(∆oX ,M ).

The construction of C+(Ψ) is to formally add an object denoted ι to

∆ × Ψ. The objects of C+(Ψ) are of the form either ι or ([n], ψ) for

n ∈ ∆ and ψ ∈ Ψ. The morphisms of C+(Ψ) are defined as follows:

—there is a single identity morphism between ι and itself;

—there are no morphisms from ([n], ψ) to ι;

—the morphisms from ([n], ψ) to ([n′], ψ′) are the same as the mor-

phisms in ∆×Ψ; and

—the morphisms from ι to ([n], ψ) are the same as the morphisms


[0] → [n] in ∆ (i.e. there are n+ 1 of them).

Composition of morphisms is defined so that the single automorphism

of ι is the identity, so that composition of morphisms within ∆ × Ψ is

the same as from that category, and a composition of the form

ι → ([n], ψ) → ([n′], ψ′)

is given by the corresponding composition [0] → [n] → [n′].

Proposition 12.7.7 Suppose M = Presh(Ψ) is a presheaf category.

There is a natural isomorphism between Presh(C+(Ψ)) and the category

of non-unital M -precategories. In particular, the latter is a presheaf cat-

egory.

Proof Suppose F : C+(Ψ)o → Set is a presheaf. Let X := F(ι). For

any (x0, . . . , xn) ∈ ∆X , let A(x0, . . . , xn) be the presheaf on Ψ which

assigns to ψ ∈ Ψ the subset of elements of F([n], ψ) which project to

(x0, . . . , xn) under the n + 1 projection maps F([n], ψ) → F(ι) = X

corresponding to the n+1 maps ι → ([n], ψ). Then pair A is an element

of Func(∆oX ,M ). Conversely, given any A ∈ Func(∆o

X ,M ), define a

presheaf F : C+(Ψ)o → Set by setting F(ι) := X and

F([n], ψ) :=∐

(x0,...,xn)∈Xn+1

A(x0, . . . , xn)(ψ).

These constructions are inverses.

The inclusion functor U∗ from PC(M ) to the category of non-unital

precategories, can be viewed as a pullback. Indeed, there is a projection

functor

π : C+(Ψ) → C(Ψ)

defined by π([n], ψ) = ([n], ψ) if n ≥ 1, π([0], ψ) = 0 and π(ι) = 0. The

pullback U∗ is just the pullback functor

π∗ : PC(M ) ∼= Presh(C(Ψ)) → Presh(C+(Ψ)).

The left adjoint of the pull back or inclusion, denoted pushforward U!

above, is also the pushforward π! for the functor π.

Using the operations C+(Ψ) and C(Ψ) we may stay essentially en-

tirely within the realm of presheaf categories. The only place where

we go outside of there is when we speak of the category of unital M -

precategories over a fixed set of objects X ; even if M is a presheaf

category, PC(X ;M ) will not generally be a presheaf category. On the

252 Precategories

other hand, the non-unital version Func(∆oX ,M ) = Presh(∆X × Ψ)

remains a presheaf category.

13

Algebraic theories in model categories

In this chapter we consider algebraic diagram theories consisting of a

collection of finite product conditions imposed on diagrams Φ → M .

This is motivated by the situation considered in the previous Chapter

12. There we defined the notion of precategory on a fixed set of objects,

which is a diagram A : ∆oX →M . The Segal conditions require that cer-

tain maps be weak equivalences. Imposing these conditions amounts to a

homotopical analogue of the “finite product theories” often considered in

category theory [146] [147], see further historical remarks in [2, pp 171-

172]. The homotopical analogue, whose origins go back to the various

theories of H-spaces and loop spaces, was considered by Badzioch in [5],

and his treatment was used by Bergner for Segal categories in [34] [39],

and also generalized to models in simplicial categories in [37]. Rosicky

carried these ideas further in [182] and he points out more references.

Let ǫ(n) denote the category

ξ0 q

q ξ1

q ξ2

q ξ3...

q ξn−1

q ξn

31:

qs

The Segal maps are obtained by pulling back A along functors ǫ(n) →

∆0X . This sets up a localization problem which can be phrased in more

general terms. We treat the general situation in the present chapter.

Even ignoring any possible other applications, that simplifies notations

for the general aspects of the problem of enforcing the given collection

of finite product conditions.


254 Algebraic theories in model categories

By an algebraic theory we mean a category Φ provided with a collection

of “direct product diagrams”, that is diagrams with the shape of a direct

product, which are functors ǫ(n)P→ Φ. A realization of the theory in a

classical 1-category C is a functor Φ → C which sends these diagrams

to direct products in C . Many of the easiest kinds of structures can

be written this way, although it is well understood that to get more

complicated structures, one needs to go to the notion of sketch which is

a category provided with more generally shaped limit diagrams. For our

purposes, it will suffice to consider direct product diagrams.

Suppose M is an appropriate kind of model category. Then a homo-

topy realization of the theory in M is a functor Φ →M which sends

the direct product diagrams, to homotopy direct products in M .

These notions lead to a “calculus of generators and relations” where

we start with an arbitrary functor Φ → C (resp. Φ →M ) and try to

enforce the direct product (resp. homotopy direct product) condition.

The main work of this chapter will be to do this for the case of homotopy

realizations in a model category M .

13.1 Diagrams over the categories ǫ(n)

The first task is to take a close look at the categories indexing direct

products. Let ǫ(n) denote the category with objects ξ0, ξ1, . . . , ξn; and

whose only morphisms apart from the identities, are single morphisms

ρi : ξ0 → ξi. We also let ξi denote the functors ξi : ∗ → ǫ(n) sending

the point ∗ to the object ξi(∗) = ξi.

This includes the case n = 0 where ǫ(0) is the discrete category with

one object ξ0.

Suppose C is some other category. A functor A : ǫ(n) → C corre-

sponds to a collection of objects a0, a1, . . . , an ∈ ob(C ) together with

maps pi : a0 → ai for 1 ≤ i ≤ n. We sometimes write

A = (a0, . . . , an; p1, . . . , pn).

Say that A is a direct product diagram if the collection of maps pi ex-

presses a0 as a direct product of the a1, . . . , an in C .

Suppose M is a model category, in particular direct products exist.

We say that a diagram A = (a0, . . . , an; p1, . . . , pn) from ǫ(n) to M is a

homotopy direct product diagram if the morphism

(p1, . . . , pn) : a0 → a1 × · · · × an



Consider now the diagram category Func(ǫ(n),M ). We describe ex-

plicitly its injective and projective model structures, which are well-

known to exist (see Hirschhorn [116], Barwick [16]).

A morphism of diagrams

A = (a0, . . . , an; p1, . . . , pn) → B = (b0, . . . , bn; q1, . . . , qn)

is a collection of maps g = (g0, . . . , gn) with gi : ai → bi such that

qig0 = gipi. We can also think of a morphism g as consisting of the

maps (g1, . . . , gn) plus a map

gP : a0 → b0 ×b1×···×bn (a1 × · · · × an)

such that the second projection is the structural map for A. We then

write g = 〈g1, . . . , gn; gP 〉.

A morphism g is fibrant in the projective structure, if and only if

each g0, . . . , gn is fibrant in M . Similarly, g is cofibrant in the injective

structure, if and only if each g0, . . . , gn is cofibrant in M .

To study fibrant maps in the injective structure, suppose

C = (c0, . . . , cn; r1, . . . , rn), D = (d0, . . . , dn; s1, . . . , sn)

are two diagrams, with morphisms forming a square

Au→ C

B

g

↓v→ D

h

↓.

These give diagrams

aiui→ ci

bi

gi

↓vi→ di

hi

↓

.

We look for a lifting f : B → C such that fg = u and hf = v. This

amounts to asking for liftings fi : bi → ci such that figi = ui and

hifi = vi, and also such that sif0 = firi.


Suppose given already the liftings f1, . . . , fn. Then to specify a full

lifting f as above we need to find f0 : b0 → c0 such that the diagram

a0 → c0

b0

g0

↓vP→

f0

→

d0 ×d1×···×dn (c1 × · · · × cn)

↓

commutes.

Lemma 13.1.1 A morphism h : C → D with notations as above, is

fibrant in the injective model structure on Func(ǫ(n),M ) if and only if

each h1, . . . , hn is fibrant in M , and the map

hP : c0 → d0 ×d1×···×dn (c1 × · · · × cn)

is fibrant in M .

Proof If h satisfies the conditions stated in the lemma, and f is a

levelwise cofibration i.e. each fi is a cofibration, then we can first choose

the liftings fi for i = 1, . . . , n, then choose f0 lifting hP . Therefore h is

fibrant. Suppose on the other hand that h is fibrant. For any cofibration

a → b and any i = 1, . . . , n we have an injective cofibration between

objects with a (resp. b) placed at i and the remaining places filled in

with ∅. Since h satisfies lifting along any such cofibration, it follows that

hi is fibrant in M . Similarly, given a cofibration a → b with a → c0 and

b → d0×d1×···×dn (c1×· · ·× cn), we get a square diagram as above with

A = (a, b, b . . . , b) and B = (b, b, . . . , b). The map A → B is a levelwise

cofibration, so the fibrant condition for h implies existence of a lifting,

which gives a lifting for the map hP .

Consider now the same question in the other direction: choose first

the lifting f0. For any i ≥ 1 we get a map b0⊔a0 ai → ci, and the choices

of fi for i = 1, . . . , n correspond to choices of lifting in the diagrams

b0 ⊔a0 ai → ci

bi

↓vi→

f i→

di

hi

↓

.

Lemma 13.1.2 A morphism g : A → B with notations as above, is

cofibrant in the projective model structure on Func(ǫ(n),M ) if and only


if g0 is cofibrant in M , and for each i = 1, . . . , n the map b0⊔a0 ai → biis cofibrant in M . In particular an object B is cofibrant if and only if b0is cofibrant and each b0 → bi is a cofibration.

Proof Similar to the previous proof.

One can describe explicit generating sets for the cofibrations and triv-

ial cofibrations in both structuresFuncinj(ǫ(n),M ) and Funcproj(ǫ(n),M ).

Recall the standard adjunctions for the functors ξi : ∗ →M . If

X ∈M is an object, we obtain a diagram ξi,!(X) : ǫ(n) →M . Explicitly,

if i = 0 then ξ0,!(X) is the constant diagram (X,X, . . . , X ; 1X, . . . , 1X)

with valuesX . If i ≥ 1 then ξ0,!(X) is the diagram (∅, . . . , ∅, X, ∅ . . . ; ι, . . . , ι)

where ι denote the unique maps from ∅ to anything else, and here X

is at the -th place. The adjunction says that for any diagram A =

(a0, . . . , an; p1, . . . , pn), a morphism ξi,!(X) → A is the same thing as a

morphism X → ξ∗i (A) = ai.

Suppose given generating sets I for the cofibrations of M and J for

the trivial cofibrations.

For f : X → Y in I, we obtain a cofibration ξ0,!(f) : ξ0,!(X) → ξ0,!(Y )

in the projective model structure. To see this, use Lemma 13.1.2 on

g = ξ0,!(f) and note that g0 is just f so it is cofibrant; and b0 ⊔a0 ai =

Y ⊔X X = Y maps to bi = Y by a cofibration. If f ∈ J then ξ0,!(f) is a

trivial cofibration: over each object of ǫ(n) we just get back the map f

so it is an levelwise weak equivalence.

For f ∈ I as above, at any i ≥ 1, ξi,!(f) is a cofibration, again using

Lemma 13.1.2 on g = ξ0,!(f). The map g0 is the identity of ∅, and the

maps b0 ⊔a0 aj → bj are either the identity of ∅ for j 6= i, or f when

j = i, so these are cofibrations. Again, if f ∈ J then ξi,!(f) is a trivial

cofibration: over each object of ǫ(n) it gives either the identity of ∅ which

is automatically a weak equivalence, or else f .

Define Iǫ(n) (resp. Jǫ(n)) to be the set consisting of diagrams of the

form ξi,!(f) for 0 ≤ i ≤ n and f ∈ I (resp. f ∈ J).

Proposition 13.1.3 The sets Iǫ(n) and Jǫ(n) are generators for the

projective model category structure Funcproj(ǫ(n),M ).

Proof By the adjunction, a morphism g in Func(ǫ(n),M ) satisfies the

right lifting property with respect to Iǫ(n) (resp. Jǫ(n)) if and only if ξ∗i (g)

satisfies the right lifting property with respect to I (resp. J) for all 0 ≤

i ≤ n. Since I (resp. J) is a set of generators for the cofibrations (resp.

trivial cofibrations) of M , this lifting property is equivalent to saying

that each ξ∗i (g) is a trivial fibration (resp. a fibration). By the definition


of the projective model structure, this is equivalent to saying that g is a

trivial fibration (resp. a fibration). Hence inj(Iǫ(n)) is the class of trivial

fibrations and inj(Jǫ(n)) is the class of fibrations, so cof(Iǫ(n)) is the class

of cofibrations and cof(Jǫ(n)) is the class of trivial cofibrations.

To get generators for the injective model structure, we need to add a

new kind of injective cofibration. If f : X → Y is a cofibration, consider

the diagrams

(f) := ξ0,!(X) ∪∐

i ξi,!(X)∐

i

ξi,!(Y ) = (X,Y, . . . , Y ; f, . . . , f) (13.1.1)

and ξ0,!(Y ). We have a map(f) → ξ0,!(Y ) which is f at the object ξ0(∗)

and 1Y at the other objects. Denote this map by ρ(f) = (f, 1, . . . , 1).

Proposition 13.1.4 Let I+ǫ(n) denote the union of Iǫ(n) with the set

of maps of the form ρ(f) = (f, 1, . . . , 1) for f ∈ I. Let J+ǫ(n) denote the

union of Jǫ(n) with the set of maps of the form ρ(f) for f ∈ J . Then I+ǫ(n)and J+

ǫ(n) are generating sets for the injective model category structure

Funcinj(ǫ(n),M ).

Proof If f : X → Y is a cofibration, then (f, 1, . . . , 1) : U → V is

a cofibration with the notations as above. Suppose that g : A → B

satisfies right lifting with respect to f , where g = (g0, . . . , gn) goes from

A = (a0, . . . , an; p1, . . . , pn) to B = (b0, . . . , bn; q1, . . . , qn). The lifting

property says that for any map u0 : X → a0 and maps ui : Y → ai for

i ≥ 1, and maps vi : Y → bi for i ≥ 0 such that piu0 = uif , vi = giuifor i ≥ 1, and v0f = g0u0, then there should exist a map u′0 : Y → a0such that u′0f = u0, g0u

′0 = v0, and piu

′0 = ui. This is the same as the

right lifting property for the square

X → a0

Y

↓→ b0 ×b1×···×bn (a1 × · · · × an)

↓

so the condition that g satisfies right lifting with respect to any (f, 1, . . . , 1)

for f ∈ J is equivalent to the condition that a0 → b0 ×b1×···×bn (a1 ×

· · · × an) is a fibration. Thus, inj(J+ǫ(n)) consists of maps which are lev-

elwise fibrations (because of lifting with respect to Jǫ(n)) and such that

the map of Lemma 13.1.1 is a fibration. By Lemma 13.1.1, inj(J+ǫ(n)) is

the class of fibrations.


Similarly, the fact that g satisfies right lifting with respect to any

(f, 1, . . . , 1) for f ∈ I is equivalent to the conditoin that

a0 → b0 ×b1×···×bn (a1 × · · · × an)

be a trivial fibration. Thus, inj(I+ǫ(n)) consists of maps which are level-

wise trivial fibrations (because of lifting with respect to Iǫ(n)) and such

that the map of Lemma 13.1.1 is a trivial fibration.

We claim that this is equal to the class of fibrations. If g ∈ inj(I+ǫ(n)),

then it is a fibration for the injective structure since J+ǫ(n) ⊂ I+ǫ(n), and

also levelwise a weak equivalence, so it is a trivial fibration. If g is a

trivial fibration, then it is levelwise a trivial fibration, and the map

a0 → b0 ×b1×···×bn (a1 × · · · × an) is a fibration. However, the maps

ai → bi are trivial fibrations, so a1× · · ·× an → b1× · · ·× bn is a trivial

fibration. Thus, the map b0 ×b1×···×bn (a1 × · · · × an) → b0 is a trivial

fibration, and by 3 for 2 we conclude that a0 → b0×b1×···×bn(a1×· · ·×an)

is a weak equivalence. Hence it is a trivial fibration as required for the

claim.

This identifies inj(I+ǫ(n)) and inj(J+ǫ(n)) with the classes of trivial fi-

brations and fibrations respectively, so cof(I+ǫ(n)) and cof(J+ǫ(n)) are the

classes of cofibrations and trivial cofibrations respectively.

Scholium 13.1.5 If (M , I, J) is a cofibrantly generated (resp. combi-

natorial, tractable, left proper) model category, then Funcinj(ǫ(n),M )

and Funcproj(ǫ(n),M ) are cofibrantly generated (resp. combinatorial,

tractable, left proper) model categories.

See Theorem 9.8.1.

13.2 Imposing the product condition

Assume that M is a tractable left proper cartesian model category.

Recall that the cartesian condition 10.0.9 implies that for any X ∈ M

and weak equivalence f : Y → Z the induced map X × Y → X × Z is

a weak equivalence.

We are going to apply the direct left Bousfield localization theory

of Chapter 11. Say that an object A ∈ Func(ǫ(n),M ) is product-

compatible if the map A(ξ0) → A(ξ1)×. . .×A(ξn) is a weak equivalence.

Let Rǫ(n) ⊂ Func(ǫ(n),M ) denote the full subcategory of product-

compatible objects.


Lemma 13.2.1 Suppose M is cartesian. The subcategory Rǫ(n) is in-

variant under weak equivalence: if A is product-compatible and its image

in hoFunc(ǫ(n),M ) is isomorphic to the image of another object B, then

B is also product-compatible.

Proof If A → B is a levelwise weak equivalence, then the horizontal

arrows in the diagram

A(ξ0) → B(ξ0)

A(ξ1)× . . .×A(ξn)

↓

→ B(ξ1)× . . .× B(ξn)

↓

are weak equivalences, using the cartesian condition for the bottom ar-

row. Hence, one of the vertical arrows is a weak equivalence if and only

if the other one is.

13.2.1 Direct localization of the projective structure

We now define the direct localizing system which goes with the full sub-

category of product-compatible objects in the projective model category

structure. For each generating cofibration f : X → Y in I, recall the

morphism ρ(f) = (f, 1, . . . , 1) defined above (see (13.1.1))

(f) = (X,Y, . . . , Y ; f, . . . , f)ρ(f)→ ξ0,!(Y ) = (Y, . . . , Y ; 1, . . . , 1).

It is obviously an injective cofibration, and the domain (f) is projec-

tively cofibrant (apply Lemma 13.1.2). However, ρ(f) will not in general

be a projective cofibration. Choose a factorization

Y ∪X Ya(f)→ Z

b(f)→ Y

such that a(f) is a cofibration and b(f) is a trivial fibration. Denoting

by i2 the first inclusion Y → Y ∪X Y we get a trivial cofibration a(f)i2 :

Y → Z. Let

(f) = (X,Y, . . . , Y ; f, . . . , f)ζ(f)→ ψ(f) := (Y, Z, . . . , Z; a(f)i2, . . . , a(f)i2)

be the map given by f over ξ0 and a(f)i1 over ξ1, . . . , ξn. Then the

map occuring in Lemma 13.1.2 is exactly a(f), so ζ(f) is a projective

cofibration.


Recall the explicit generating set Jǫ(n) for the trivial cofibrations in

the projective model structure. Put

Kprojǫ(n) := Jǫ(n) ∪ ζ(f)f∈I .

Theorem 13.2.2 The pair (Rǫ(n),Kprojǫ(n)) is a direct localizing system

for the model category Funcproj(ǫ(n),M ) of ǫ(n)-diagrams in M with

the projective model structure. Let Funcproj,Π(ǫ(n),M ) denote the left

Bousfield localized model structure constructed in Chapter 11. A mor-

phism A → B is a weak equivalence if and only if it induces weak equiv-

alences levelwise over the objects ξ1, . . . , ξn. An object A is fibrant in the

localized structure if and only if it is product-compatible and each A(ξi)

is fibrant.

Proof We verify the properties (A1)–(A6). Properties (A1) and (A2)

are immediate. For (A3) we have chosen ζ(f) so as to be a projectively

cofibration, and its domain is projectively cofibrant. Condition (A4) is

given by Lemma 13.2.1.

Suppose a diagramA is in inj(Kprojǫ(n)). In particular it is Jǫ(n)-injective,

which is to say fibrant in the projective model structure. This means that

each A(ξi) is a fibrant object in M . The lifting property along the ζ(f)

implies the following homotopy lifting property of the map

p : A(ξ0) → A(ξ1)× · · · × A(ξn).

along a generating cofibration f : X → Y in I. Recall that Y ∪X

Ya→ Z

b→ Y was chosen above. If we are given a diagram

Xu

→ A(ξ0)

Y

f

↓v→ A(ξ1)× · · · × A(ξn)

p

↓

then there is a map Yw→ A(ξ0) such that wf = u, and a map Z

h→ A(ξ1)×

· · · × A(ξn) such that hai2 = v but hai1 = pw. This homotopy lifting

property implies that p is a weak equivalence, see Lemma 9.4.1. Thus,

A ∈ Rǫ(n) which gives (A5).

For condition (A6), suppose A is in Rǫ(n) and A → B is a pushout

along an element of Kprojǫ(n) . Applying the small object argument we can

find a map B → C in cell(Kprojǫ(n)) such that C ∈ inj(Kproj

ǫ(n)). Then A → C


is also in cell(Kprojǫ(n)). Notice, however, that the elements of Kproj

ǫ(n) are

levelwise trivial cofibrations over the objects ξ1, . . . , ξn ∈ ǫ(n) (but not

over ξ0). Therefore the map A → C induces weak equivalences over each

ξ1, . . . , ξn. Using the cartesian condition on M , this implies that the

right vertical map in the square

A(ξ0) → A(ξ1)× · · · × A(ξn)

C(ξ0)

↓

→ C(ξ1)× · · · × C(ξn)

↓

is a weak equivalence. The hypothesis that A ∈ Rǫ(n) says that the top

map is a weak equivalence, and the fact that C ∈ inj(Kprojǫ(n)) and part

(A5) proved above say that C ∈ Rǫ(n), so the bottom map is a weak

equivalence. By 3 for 2 the left vertical arrow is a weak equivalence,

showing that A → C is a levelwise weak equivalence of diagrams. This

shows (A6).

Our direct localizing system leads to a left Bousfield localization by

Theorem 11.7.1.

We now look at the characterizations of new weak equivalences. As

seen above, the elements of cell(Kprojǫ(n)) are weak equivalences levelwise

over the ξ1, . . . , ξn. Using the characterizatin of Corollary 11.4.3 we see

that all new weak equivalences are levelwise weak equivalences over the

ξ1, . . . , ξn. Suppose Af→ B is a morphism inducing a weak equivalence

over each ξ1, . . . , ξn. Choose Bb→ B′ in cell(Kproj

ǫ(n)) such that B′ is in

inj(Kprojǫ(n)), in particular it is product-compatible. Factor the composed

map as

Aa→ A′ g

→ B′

where a ∈ cell(Kprojǫ(n)) and g ∈ inj(Kproj

ǫ(n)). All of the above maps are

levelwise weak equivalences over the objects ξ1, . . . , ξn, however A′ and

B′ are product-compatible. It follows that g is a levelwise weak equiv-

alence. The criterion of Corollary 11.4.3 implies that f is a new weak

equivalence.

The characterization of fibrant objects is a first version in the sim-

plified situation of a single product diagram, of Bergner’s characteri-

zation of fibrant Segal categories [36]. The new fibrant objects are in

inj(Kprojǫ(n)) so they are in Rǫ(n) i.e. product-compatible, and levelwise

fibrant. Suppose A is product-compatible and levelwise fibrant. Suppose


Uf→ V is a new trivial cofibration, and we are given a map U

u→ A.

By the previous paragraph it induces a levelwise trivial cofibration over

the objects ξ1, . . . , ξn. Hence the components u1, . . . , un extend to maps

V(ξi)v′i→ A(ξi). Putting these together, the composition

V(ξ0) → V(ξ1)× · · · × V(ξn) → A(ξ1)× · · · × A(ξn)

gives the bottom arrow of the diagram

U(ξ0)u0

→ A(ξ0)

V(ξ0)

f0

↓

→ A(ξ1)× · · · × A(ξn).

↓

The right vertical arrow is a weak equivalence between fibrant objects,

so by Lemma 9.4.1 there is a homotopy lifting relative U(ξ0), in other

words a map V(ξ0)v0→ A(ξ0) such that rf0 = u0, and the other triangle

commutes up to a homotopy relative U(ξ0). Lemma 9.4.2 says we can

change the maps v′i to maps vi, still restricting to ui on U(ξi), but

compatible with v0. We have now constructed the required extension

V → A, showing that A is a new fibant object.

13.2.2 Direct localization of the injective structure

When possible, it is more convenient to use the injective model structure

on Func(ǫ(n),M ). Consider the explicit generating set J+ǫ(n) for the

trivial cofibrations in the injective model structure, given by Proposition

13.1.4.We can define two different sets of cofibrations, the first extending

Kprojǫ(n) :

K inj+ǫ(n) := Kproj

ǫ(n) ∪ J+ǫ(n) = J+

ǫ(n) ∪ ζ(f)f∈I ;

and the second defined using the simpler maps ρ(f) which were already

injective cofibrations:

K injǫ(n) := J+

ǫ(n) ∪ ρ(f)f∈I .

Theorem 13.2.3 The pairs (Rǫ(n),Kinj+ǫ(n) ) and (Rǫ(n),K

injǫ(n)) are both

direct localizing systems for the model category Funcinj(ǫ(n),M ) of

ǫ(n)-diagrams in M with the injective model structure. Let Funcinj,Π(ǫ(n),M )

denote the left Bousfield localized model structure, which is the same in


both cases. The weak equivalences are the same as for the projective

structure. An object A is fibrant in the localized structure if and only

if it is product-compatible and satisfies the fibrancy criterion of Lemma

13.1.1 for the injective model structure. The identity functor is a left

Quillen functor

Funcproj,Π(ǫ(n),M ) → Funcinj,Π(ǫ(n),M )

from the new projective to the new injective model structure.

Proof The functor Funcproj(ǫ(n),M ) → Funcinj(ǫ(n),M ) is a left

Quillen functor, whose corresponding right Quillen functor (both be-

ing the identity on underlying categories) preserves the class Rǫ(n) of

product-compatible diagrams. In other words the transfered class is the

same. The subsetK inj+ǫ(n) is the transfered subset given in Theorem 11.8.1,

so by that theorem (Rǫ(n),Kinj+ǫ(n) ) is a direct localizing system and the

identity functor is a left Quillen functor from the previous new projec-

tive model structure to the resulting left Bousfield localization of the

injective structure

Funcproj,Π(ǫ(n),M ) → Funcinj,Π(ǫ(n),M ).

The proof that (Rǫ(n),Kinjǫ(n)) is a direct localizing system is the same

as in the proof of the previous Theorem 13.2.2, but in fact easier since

an object which satisfies lifting with respect to the ρ(f) has the stronger

property that

A(ξ0) → A(ξ1)× · · · × A(ξn)

is in inj(I), that is it is a trivial fibration. So in this case we don’t

need to rely on the notion of homotopy lifting property as was done

in the previous proof. We get conditions (A1)–(A6) and also the same

description of weak equivalences, and the corresponding description of

fibrant objects.

The two model structures given by Theorem 11.7.1 applied to (Rǫ(n),Kinjǫ(n))

and (Rǫ(n),Kinj+ǫ(n) ) are the same, by Proposition 11.7.2.

Suppose A ∈ Func(ǫ(n),M ). Suppose given a factorization

A(ξ0)e0→ E0

p→ A(ξ1)× · · · × A(ξn)

in M . Let Ei := A(ξi) for i = 1, . . . , n. The structural map p gives a

structure of ǫ(n)-diagram to the collection (E0, . . . , En), call it E . The

map e gives a map e : A → E . If e0 is a cofibration in M then e is a

cofibration in Funcinj(ǫ(n),M ).


Lemma 13.2.4 In the above situation, if e0 is a cofibration and p is

a weak equivalence in M then e : A → E is a trivial cofibration in

Funcinj,Π(ǫ(n),M ).

Proof The map e is levelwise cofibrant by construction. It is a weak

equivalence since it induces a weak equivalence levelwise over the objects

ξ1, . . . , ξn.

13.2.3 Transfering these structures

Putting together the above analysis of diagrams over ǫ(n) with the

transfer along a Quillen functor gives the following general picture.

Suppose we are given a set Q, integers n(q) ≥ 0 for q ∈ Q, a fam-

ily of tractable left proper cartesian model categories Mq for q ∈ Q, a

tractable left proper model category N , and a family of Quillen functors

Fq : Funcproj(ǫ(n(q)),Mq) ←→ N : Gq.

Let R′ ⊂ N be the full subcategory of objects Y such that, for a

fibrant replacement Y → Y ′, the diagrams Gq(Y′) : ǫ(n(q)) →Mq are

product-compatible. Let (Iq, Jq) be generating sets for Mq, and (I ′, J ′)

generators for N .

Corollary 13.2.5 Let K ′ be the union of J ′, of the set of morphisms

of the form Fq(g) for g ∈ Jqǫ(n(q)), and of the set of morphisms of the

form Fq(ζ(f)) for f ∈ Iq. Then (R′,K ′) is a direct localizing system for

N .

If furthermore Fq are left Quillen functors from Funcproj(ǫ(n(q)),Mq)

to N , then we can consider K inj, the union of J ′ with the set of mor-

phisms of the form Fq(ρ(f)) for f ∈ Iq, and (R′,K inj) is a direct local-

izing system for N giving the same model structure as (R′,K ′).

Proof Let K ′′ be the union of K ′ with the set of morphisms of the form

Fq(g) for g ∈ Jqǫ(n(q)). Then (R′,K ′′) is a direct localizing system for N ,

by Theorem 11.8.3 applied to the direct localizing systems of Theorem

13.2.2. However, the Fq(g) for g ∈ Jqǫ(n(q)) are trivial cofibrations between

cofibrant objects in the original model structure of N , so they could be

included in a bigger generating set J ′′ for the original trivial cofibrations

of N . But one can note that in the construction of a direct localizing

system by adding on some new morphisms to the original generating set,

the properties are independent of the choice of original generating set.

So K ′ works as well as K ′′.

Suppose now that Fq remain left Quillen functors when we use the in-

jective model structures on their sources. Then, with a similar discussion


for leaving out the images of the morphisms in Jq,+ǫ(n(q)), Theorem 11.8.3

applies to the direct localizing systems of Theorem 13.2.3 to conclude

that (R′,K inj) is a direct localizing system. As pointed out in Proposi-

tion 11.7.2, the resulting model structure is the same as for (R′,K ′).

13.3 Algebraic diagram theories

Classically, an “algebraic theory” is given by a small category Φ and a

collection of product diagrams Pq : ǫ(n(q)) → Φ. The objects of the

theory are the functors A : Φ → Set with the property that p∗q(A) is

a direct product, that is product-compatible in the above terminology.

Of course this theory has since been much generalized, to include the

notion of “finite limit sketches” among other things. However, for our

purposes it will be sufficient to consider just the basic version of the

theory, and to give it a weak-enriched counterpart using the notion of

direct left Bousfield localization we have developped so far.

So, suppose Φ is a small category, Q is a small set, we have integers

n(q) ≥ 0 for q ∈ Q, and suppose given functors Pq : ǫ(n(q)) → Φ for

q ∈ Q.

For the coefficients, fix a tractable left proper model category M

satisfying condition (PROD). Let (I, J) be a set of generators for M .

Playing the role of the model category N will be the category of Φ-

diagrams in M with its projective or injective model structure, N =

Funcproj(Φ,M ) (resp. N = Funcproj(Φ,M )). Let (IΦ,proj, JΦ,proj) and

(IΦ,inj, JΦ,inj) be the sets of generators for the projective and injective

model structures respectively, see the discussion of references for The-

orem 9.8.1. Recall that the identity functor is a Quillen adjunction be-

tween the projective and injective diagram categories

1 : Funcproj(Φ,M ) ←→ Funcinj(Φ,M ) : 1.

Lemma 13.3.1 With the above notations, for each q ∈ Q we get a

Quillen adjunction

Pq,! : Funcproj(ǫ(n(q)),M ) ←→ Funcproj(Φ,M ) : P ∗q .

This composes with the identity functor to give a Quillen adjunction

1Pq,! : Funcproj(ǫ(n(q)),M ) ←→ Funcinj(Φ,M ) : P ∗q 1.

Proof This is the standard Quillen adjunction between Bousfield pro-

jective model structures coming from the functor Pq : ǫ(n(q)) → Φ.

13.3 Algebraic diagram theories 267

In the above situation, let R(Φ, P·,M ) denote the full subcategory of

Func(Φ,M ) consisting of diagrams A such that for a fibrant replace-

ment A → A′ in the injective model structure (which is also a fibrant

replacement in the projective model structure), for all q ∈ Q, P ∗q (A

′) ∈

Func(ǫ(n(q)),M ) is product-compatible. Let Kproj/inj(Φ, P·,M , I, J)

be the sets given by Corollary 13.2.5 for the projective/injective struc-

ture, consisting of the elements of JΦ,proj/inj, of the Pq,!(Jǫ(n(q))), and of

the Pq,!(ρǫ(n(q))(f)) for f ∈ I. Here a choice of subscript proj or inj is

proposed when necessary.

Theorem 13.3.2 The pair (R(Φ, P·,M ),Kproj/inj(Φ, P·,M , I, J)) is

a direct localizing system. Define the model category of weak (Φ, P·)-

algebras in M denoted by Algproj/inj(Φ, P·;M ), to be the direct left

Bousfield localization of the projective or injective diagram model cate-

gory Funcproj/inj(Φ,M ) with respect to the full subcategory R(Φ, P·,M )

and sets Kproj/inj(Φ, P·,M , I, J). The cofibrations are levelwise cofibra-

tions in the injective structure, and projective diagram cofibrations in

the projective structure. The fibrant objects of Algproj/inj(Φ, P·;M ) are

the diagrams A : Φ →M such that A is levelwise fibrant (for the pro-

jective structure) or fibrant in the injective diagram structure, and for

each q ∈ Q the pullback P ∗q (A) : ǫ(n(q)) →M is product-compatible.

Given a map f : A → B of Φ-diagrams in M , the following conditions

are equivalent:

—f is a weak equivalence in Algproj/inj(Φ, P·;M )

—for any square diagram

A ← A′ → A′′

B↓← B′

↓

→ B′′

↓

such that the left horizontal arrows are projective/levelwise cofibrant re-

placements, and A′′,B′′ ∈ R(Φ, P·,M ), then the morphism A′′ → B′′

is an levelwise weak equivalence;

—there exists a square diagram as above with A′′ → B′′ an levelwise

weak equivalence;

—there exists a square diagram as above with A′′ → B′′ an levelwise

weak equivalence, but without the requirement A′′,B′′ ∈ R(Φ, P·,M ).

These projective and injective model categories of weak algebras are

tractable and left proper.


Proof Apply the construction of the direct left Bousfield localization

given in the previous chapter, starting with either Funcproj(Φ,M ) or

Funcinj(Φ,M ) of Theorem 9.8.1, and transfering the direct localizing

systems as in Corollary 13.2.5.

For the characterization of fibrant objects, see the successive state-

ments of Proposition 9.9.8, Theorem 11.7.1, Remark 11.8.2 and Theo-

rem 11.8.3. Apply these together with the characterizations of fibrant

objects in the product-compatible ǫ(n)-diagram model structures.

13.4 Unitality

Suppose given a full subcategory Φ0 ⊂ Φ. Typically, these will be the

Pq(ξ0) for q ∈ Q such that n(q) = 0. We would like to consider diagrams

A : Φ →M such that A(x) = ∗ for x ∈ Φ0. Call such a diagram unital

along Φ0.

Let Func(Φ/Φ0,M ) ⊂ Func(Φ,M ) denote the full subcategory of

diagrams which are unital along Φ0. Denote by

U∗Φ0

: Func(Φ/Φ0,M ) → Func(Φ,M )

the identity inclusion functor.

The idea for this notation is that Φ/Φ0 represents the contraction of

Φ0 to a point, the result being a pointed category i.e. a category with

distinguished object; and Func(Φ/Φ0,M ) is the category of pointed

functors from here to the category M pointed by distinguishing the

coinitial object ∗.

The present discussion plays an important role in the theory of weakly

enriched precategories: the unitality condition corresponds to Tamsamani’s

constancy condition in the case of n-precategories, corresponding to the

idea of having a globular theory in which the objects form a discrete set.

The motivating example, first introduced in Section 12.1 above, is when

Φ = ∆X and Φ0 = X is the subcategory of sequences of length 0.

For any x ∈ Φ denote by Φ0/x the category of arrows z → x with

z ∈ Φ0.

Theorem 13.4.1 If M is a locally presentable category, then the cat-

egory Func(Φ/Φ0,M ) is locally presentable and U∗Φ0

has a left adjoint

UΦ0,!. The left adjoint is given as follows: if A ∈ Func(Φ,M ) then

UΦ0,!A is the diagram which sends an object x ∈ Φ to the coprod-

uct of A(x) and colimΦ0/x∗ over colimz∈Φ0/xA(z). The adjunction map

13.4 Unitality 269

UΦ0,!U∗Φ0B → B is the identity for any B ∈ Func(Φ/Φ0,M ), so UΦ0,!

is a monadic projection in the terminology of Section 8.2.

The full subcategory Func(Φ/Φ0,M ) ⊂ Func(Φ,M ) is closed un-

der small limits and over colimits with small nonempty connected index

sets, in particular it is closed under coproducts, filtered colimits, and

transfinite composition.

For any regular cardinal κ with M being locally κ-presentable and

|Φ| < κ, an object A ∈ Func(Φ/Φ0,M ) is κ-presentable if and only if

each of the A(x) are κ-presentable in M .

Proof Put

UΦ0,!(A)(x) := A(x) ∪colimz∈Φ0/xA(z) colimΦ0/x ∗ .

Given a morphism x → y, we obtain morphisms

colimΦ0/x ∗ → colimΦ0/y∗

and

colimz∈Φ0/xA(z) → colimz∈Φ0/yA(z).

These are compatible with the maps in the above coproduct so they give

UΦ0,!(A) a structure of diagram (i.e. functor). If x ∈ Φ0 then x is the

coinitial object of Φ0/x, and we get UΦ0,!(A)(x) = A(x) ∪A(x) ∗ = ∗.

Thus UΦ0,!(A) ∈ Func(Φ/Φ0,M ).

To show adjunction, suppose B ∈ Func(Φ/Φ0,M ). Given a map

A → U∗Φ0B then for any z ∈ Φ0 the map A(z) → B(z) factors through

∗ (since indeed B(z) = ∗). For any x this gives a factorization

colimz∈Φ0/xA(z) → colimz∈Φ0/x∗

A(x)

↓

→ B(x)

↓

so our map of diagrams factors through a unique map UΦ0,!(A) → B.

If A = U∗Φ0B is a diagram with A(z) = ∗ for z ∈ Φ0 already, then

the second map in the coproduct defining UΦ0,!(A) is the identity, so

UΦ0,!(A) = A, i.e. the adjunction is a monadic projection.

Closure under arbitrary small limits is automatic since U∗Φ0

is a right

adjoint. For closure under connected colimits, suppose α is an index

category with connected nerve. Then colimα∗ = ∗ where ∗ is the coini-

tial object of M and the colimit is taken over the constant functor


α →M (Lemma 8.1.8). As colimits in Func(Φ,M ) are calculated

levelwise, it follows that colimits over α preserve the condition for in-

clusion in Func(Φ/Φ0,M ) which is that the diagram take values ∗ lev-

elwise over Φ0. Note that this property says that connected colimits in

Func(Φ/Φ0,M ) are calculated levelwise. That wouldn’t be true, how-

ever, for disconnected colimits such as disjoint sums.

We now identify the κ-presentable objects of Func(Φ/Φ0,M ). If each

A(x) is a κ-presentable object of M , then by Lemma 8.1.3 A is κ-

presentable in Func(Φ,M ). If we are given a κ-filtered system Bii∈βin Func(Φ/Φ0,M ), any map

A → colimFunc(Φ/Φ0M )i∈β Bi

is also a map to colimFunc(Φ,M )i∈β Bi by the closure under connected

colimits; hence it factors through one of the Bi. This factorization is

a morphism in the full subcategory Func(Φ/Φ0,M ), which shows that

A is κ-presentable in Func(Φ/Φ0,M ).

Suppose on the other hand thatA is κ-presentable in Func(Φ/Φ0,M ).

Given our assumption that M is locally κ-presentable and |Φ| < κ, the

category Func(Φ,M ) is locally κ-presentable, and its κ-presentable ob-

jects are exactly the diagrams B such that B(x) is κ-presentable in M .

This was stated as Lemma 8.1.3 with reference to [2]. In particular, we

can express A as a colimit in the category Func(Φ,M )

A = colimi∈βBi

with Bi(x) being κ-presentable, and indexed by a κ-filtered category β.

The unitalization functor UΦ0,! being a left adjoint, we get

A = colimi∈βUΦ0,!(Bi) in Func(Φ/Φ0M ). (13.4.1)

The hypothesis that A is κ-presentable in Func(Φ/Φ0,M ), applied to

the identity map of A, says that the identity factors through a map

A → UΦ0,!(Bi). On the other hand, the explicit description of UΦ0,!

shows that each UΦ0,!(Bi)(x) is κ-presentable. We have a retraction

A(x) → UΦ0,!(Bi)(x) → A(x)

the composition being the identity of A(x). It easily follows that A(x)

is κ-presentable in M . This completes the proof of the identification of

κ-presentable objects of Func(Φ/Φ0M ).

It is clear from this description that the κ-presentable objects form

a small set. The above argument shows that any A ∈ Func(Φ/Φ0M )

13.4 Unitality 271

is a κ-filtered colimit of κ-presentable objects, indeed we obtained the

expression (13.4.1).

We can construct the injective model category structure.

Proposition 13.4.2 Suppose M is tractable. There exists a tractable

injective model category structure Funcinj(Φ/Φ0,M ) where the weak

equivalences are levelwise weak equivalences, and cofibrations are lev-

elwise cofibrations. If M is left proper then so is the model category

Funcinj(Φ/Φ0,M ).

Proof Weak equivalences are clearly closed under retracts and satisfy

3 for 2. The class of trivial cofibrations, that is the intersection of the

classes of cofibrations and weak equivalences, is closed under pushout

and transfinite composition since these colimits are calculated levelwise.

The sets of injective cofibrations and injective trivial cofibrations have

generating sets, as was shown in Theorem 8.9.3 using Lurie’s technique of

Theorems 8.9.1 and 8.9.2. This gives the necessary accessibility argument

which allows to apply Smith’s recognition lemma to obtain the model

structure, such as described in [16]. If M is left proper, the colimits

involved in this condition are connected so they are computed levelwise,

hence the same condition holds for Funcinj(Φ/Φ0,M ).

And the projective structure.

Proposition 13.4.3 If Φ0 ⊂ Φ and M is a tractable left proper model

category, we get a projective model structure Funcproj(Φ/Φ0,M ) which

is a tractable left proper model category. Furthermore the identity on the

underlying category constitutes a left Quillen functor

Funcproj(Φ/Φ0,M ) → Funcinj(Φ/Φ0,M ),

and the unitalization construction is a left Quillen functor

Funcproj(Φ,M )UΦ0,!→ Funcproj(Φ/Φ0,M ).

Proof The weak equivalences in Funcproj(Φ/Φ0,M ) are defined to be

the levelwise weak equivalences. These satisfy 3 for 2 and are closed

under retracts. The fibrations are defined to be the levelwise fibrations.

The trivial fibrations are the intersection of these classes. The cofibra-

tions are determined by the left lifting property with respect to trivial

fibrations. We construct explicitly a generating set, by a small variant

of Bousfield’s original construction.

Choose a generating set I for the cofibrations of M . It leads to the


set IΦ of generators for cofibrations in the projective model structure

Funcproj(Φ,M ) discussed in Theorem 9.8.1. Recall that IΦ consists of

all morphisms of the form ix,!(f) where f : A → B is in I, where

ix : x → Φ is the inclusion of a discrete single object and where

ix,! : M = Func(x,M ) → Func(Φ,M )

is the corresponding left adjoint functor. This is just Bousfield’s classic

generating set for projective diagram cofibrations. Set

IΦ/Φ0:= UΦ0,!(IΦ).

Notice that

UΦ0,!ix,! : M = Func(x,M ) → Func(Φ/Φ0,M )

is the left adjoint functor for inducing unital diagrams from objects of

M placed over x ∈ Ob(Φ). The set IΦ/Φ0consists of all UΦ0,!ix,!(f)

where f runs through the set I of generating cofibrations for M and x

runs through Ob(Φ).

For x and f : A → B fixed,

UΦ0,!ix,!(f) : UΦ0,!ix,!(A) → UΦ0,!ix,!(B)

has the following explicit description. For any object y ∈ Φ, let Φnf(x, y)

denote the set of arrows from x to y which don’t factor through an

objects of Φ0, and let Φf(x, y) denote the set of arrows which factor

through an element of Φ0. Thus Φ(x, y) = Φnf(x, y) ⊔Φf(x, y). Then,

UΦ0,!ix,!(A)(y) =∐

Φnf (x,y)

A, if Φf(x, y) = ∅;

UΦ0,!ix,!(A)(y) = ∗ ⊔∐

Φnf (x,y)

A, if Φf(x, y) 6= ∅.

For an arrow y → z, composition induces Φf(x, y) → Φ(x, z) but only

Φnf(x, y) → Φnf(x, y)⊔Φf(x, y). The morphisms of functoriality for the

diagram UΦ0,!ix,!(A) are either the identity on A or on ∗, or else the

projection A → ∗ in the case of an arrow in Φnf(x, y) which composes

with y → z to give an arrow in Φf(x, z).

Note that if u ∈ Φ0 then Φnf(x, u) = ∅ and UΦ0,!ix,!(A)(u) = ∗ so

the above formula defines a unital diagram. One can check by hand

that the explicit construction described above is adjoint to the functor

Func(Φ/Φ0,M ) →M of evaluation at x, which serves to show that

the explicit construction is indeed UΦ0,!ix,!.

13.4 Unitality 273

The same description holds for UΦ0,!ix,!(B) and the map UΦ0,!ix,!(f)

is obtained by applying either f or 1∗ on the various factors.

A few things are immediate from this description:

(1) if f is any cofibration in M then UΦ0,!ix,!(f) is an injective i.e.

levelwise cofibration in Func(Φ/Φ0;M ), indeed the UΦ0,!ix,!(f)(y) are

disjoint unions of copies of f and of the isomorphism 1∗; and

(2) if f is a trivial cofibration in M then UΦ0,!ix,!(f) is an injective i.e.

levelwise trivial cofibration in Func(Φ/Φ0;M ), for the same reason.

On the other hand, the adjunction formula says that UΦ0,!ix,! is left

adjoint to the restriction

i∗x : Func(Φ/Φ0,M ) →M

i.e. the evaluation at x ∈ Φ. Hence, a morphism g of diagrams in

Func(Φ/Φ0,M ) satisfies right lifting with respect to UΦ0,!ix,!(f), if and

only if i∗x(g) = g(x) satisfies the right lifting property with respect to f .

The previous paragraph implies that inj(IΦ/Φ0) is equal to the class of

levelwise trivial fibrations, hence cof(IΦ/Φ0) is the class of cofibrations.

Thus IΦ/Φ0is a set of generators for the cofibrations.

If we had started with a set J generating the trivial cofibrations of M ,

then defining JΦ/Φ0to be the set of all UΦ0,!ix,!(f) for f ∈ J , gives by

the same argument inj(JΦ/Φ0) equal to the class of levelwise fibrations.

We claim that cof(JΦ/Φ0) is then equal to the class of trivial cofibra-

tions. By property (2) above, JΦ/Φ0and hence cof(JΦ/Φ0

) consist of

levelwise weak equivalences, so they are contained in the class of trivial

cofibrations. Suppose g : R → S is a trivial cofibration. By the small

object argument it can be factored as g = ph where p ∈ inj(JΦ/Φ0)

and h ∈ cell(JΦ/Φ0). In particular p is an levelwise fibration, but it is

also an levelwise weak equivalence by 3 for 2, so it is a trivial fibration

hence satisfies lifting with respect to cofibrations. As g is assumed to

be a cofibration, there is a lifting which shows g to be a retract of h.

Thus g ∈ cof (JΦ/Φ0). We have now shown conditions (CG1)–(CG3b)

for I and J with respect to the given three classes of morphisms, and

we know that weak equivalences are closed under retracts and satisfy 3

for 2. These give a cofibrantly generated model structure (see Proposi-

tion 9.2.1). It is tractable since the elements of the generating sets have

cofibrant domains. Left properness is checked levelwise.

For the statements about left Quillen functors, note that both functors

in question are left adjoints. Furthermore, they preserve cofibrations and

trivial cofibrations, indeed by (1) and (2) above the generating cofibra-

tions of Funcproj(Φ/Φ0,M ) are also injective cofibrations; and by our


construction the generating sets IΦ and JΦ for cofibrations and trivial

cofibrations in Funcproj(Φ,M ) are mapped by UΦ0,! to the generating

sets for cofibrations and trivial cofibrations in Funcproj(Φ/Φ0,M ).

Remark 13.4.4 Unfortunately U! is not necessarily a left Quillen

functor between the injective structures.

In the next chapter when our discussion is applied to the special case

∆oX/X we can impose an additional condition 14.2.2 on M so that

it works, or alternatively use the Reedy structure on the category of

diagrams.

13.5 Unital algebraic diagram theories

Combine the previous discussions: suppose Φ is a small category, Φ0 ⊂ Φ

is a full subcategory,Q is a small set, we have integers n(q) ≥ 0 for q ∈ Q,

and suppose given functors Pq : ǫ(n(q)) → Φ for q ∈ Q. Suppose that

M is a tractable left proper cartesian model category with generating

sets I and J . We obtain left Quillen functors (the leftmost varying in a

family indexed by q ∈ Q):

Funcproj(ǫ(n(q)),M )Pq,!→ Funcproj(Φ,M )

Funcproj(Φ/Φ0,M )

UΦ0,!

↓1→ Funcinj(Φ/Φ0,M ).

By the property of transfer of families of direct localizing systems along

Quillen functors (Theorem 11.8.3), we obtain left Bousfield localizations

of Funcproj(Φ/Φ0,M ) and Funcinj(Φ/Φ0,M ) along the images of the

ζǫ(n(q))(f) for f ∈ I. Denote these respectively by Algproj(Φ/Φ0, P·;M )

and Alginj(Φ/Φ0, P·;M ). They are tractable left proper model cat-

egories whose underlying categories are the unital diagram category

Func(Φ/Φ0,M ). In the projective structure, the cofibrations are gener-

ated by UΦ0,!(IΦ) whereas in the injective structure the cofibrations are

the levelwise cofibrations. The fibrant objects in the projective structure

are the levelwise fibrant objects whose pullback to each ǫ(n(q)) satisfies

the product condition.

In the projective case this compares with the non-unital algebraic

13.6 The generation operation 275

diagram theories by a Quillen adjunction

UΦ0,! : Algproj(Φ, P·;M ) ←→ Algproj(Φ/Φ0, P·;M ) : U∗Φ0.

Indeed, the direct left Bousfield localization of the unital theory can be

seen as coming from the localization of the non-unital theory given in

Theorem 13.3.2, by transfer along the left Quillen functor UΦ0,!, and in

the situation of Theorem 11.8.1 we still get a left Quillen functor.

Remark 13.5.1 If furthermore we know that UΦ0,! gives a left Quillen

functor on the injective diagram structures, then this completes to a

Quillen adjunction

UΦ0,! : Alginj(Φ, P·;M ) ←→ Alginj(Φ/Φ0, P·;M ) : U∗Φ0.

Lemma 13.5.2 Suppose r : A → A′ is a trivial cofibration towards

a fibrant object, in either of the projective or injective model structures

on Alg(Φ/Φ0, P·;M ). If A satisfies the product condition, then r is

levelwise a weak equivalence, that is r(x) : A(x) → A′(x) is a weak

equivalence in M for any x ∈ Φ.

Proof This follows from Corollary 11.4.5, noting that the model struc-

tures on Alg(Φ/Φ0, P·;M ) are obtained from direct left Bousfield lo-

calizing systems with R being the class of objects satisfying the product

condition.

13.6 The generation operation

Suppose Φ is a small category, Φ0 ⊂ Φ is a full subcategory, Q is a small

set, we have integers n(q) ≥ 0 for q ∈ Q, and suppose given functors

Pq : ǫ(n(q)) → Φ for q ∈ Q. Suppose that M is a tractable left proper

model category with generating sets I and J .

make the following assumption:

(INJ)—the functors UΦ0,!Pq,! send cofibrations (resp. trivial cofibrations)

in Funcinj(ǫ(nq),M ) to levelwise cofibrations (resp. levelwise trivial

cofibrations).

In other words we have left Quillen functors

Funcinj(ǫ(nq),M )UΦ0,!Pq,!

→ diaginj(Φ/Φ0,M ).

In this case we can use the generating set for the new model structure

on Funcinj(ǫ(nq),M ) made from the simpler cofibrations ρ(f).


Suppose A ∈ Func(Φ/Φ0,M ) and q ∈ Q. Let n := nq. Define a

trivial cofibration A → Gen(A; q) as follows: choose a factorization

P ∗q (A)(ξ0)

e0→ E0

p→ P ∗

q (A)(ξ1)× · · · × P∗q (A)(ξn)

with e0 a cofibration and p a weak equivalence, in M . This gives a trivial

cofibration P ∗q (A)

e→ E in Funcinj,Π(ǫ(n),M ). Using condition (INJ)

we obtain a cofibration

A → Gen(A; q) := A∪UΦ0,!Pq,!(A) UΦ0,!Pq,!(E) (13.6.1)

in the injective model structure Funcinj(Φ/Φ0,M ).

Note that Gen(A, q) doesn’t depend, up to equivalence, on the choice

of factorization E. If necessary we can include the factorization in the

notation Gen(A, q; e0, p).

The weak monadic projection from Func(Φ/Φ0,M ) to the class of

objects satisfying the product condition, may be thought of as a trans-

finite iteration of the operation A 7→ Gen(A, q) over all q ∈ Q.

13.7 Reedy structures

In the main situation where the theory of this chapter will be applied, the

underlying category Φ is a Reedy category, and we can give the category

of diagrams Φ →M the Reedy model category structure denoted by

FuncReedy(Φ,M ).

Assume that Φ0 consists of objects of the bottom degree in the Reedy

structure. Then there is a corresponding model structure denoted FuncReedy(Φ/Φ0,M )

on the category of unital diagrams, such that (U!, U∗) remains a Quillen

adjunction. The Reedy structuresFuncReedy(Φ,M ) and FuncReedy(Φ/Φ0,M )

can again by localized by direct left Bousfield localization, to give model

categories denotedAlgReedy(Φ, P·;M ) andAlgReedy(Φ/Φ0, P·;M ). These

fit in between the projective and the injective structures above.

14

Weak equivalences

This chapter continues the study of weakly enriched categories using

Segal’s method. We use the model category for algebraic theories, devel-

opped in the previous chapter, to get model structures for Segal precat-

egories on a fixed set of objects. This structure will be studied in detail

later, to deal with the passage from a Segal precategory to the Segal

category it generates.

Then we consider the full category of Segal precategories, with mov-

able sets of objects, giving various definitions and notations. Construct-

ing a model structure in this case is the main subject of the subsequent

chapters.

The reader will note that this division of the global argument into

two pieces, was present already in Dwyer-Kan’s treatment of the model

category for simplicial categories. They discussed the model category

for simplicial categories on a fixed set of objects in a series of papers

[89] [90] [91]; but it wasn’t until some time later with their unpublished

manuscript with Hirschhorn [87], which subsequently became [88], and

then Bergner’s paper [33] that the global case was treated.

For the theory of weak enrichment following Segal’s method, the cor-

responding division and introduction of the notion of left Bousfield lo-

calization for the first part, was suggested in Barwick’s thesis [14].

Assume throughout that M is a tractable left proper cartesian model

category. See Chapter 10 for an explanation and first consequences of

the cartesian condition.


278 Weak equivalences

14.1 The model structures on PC(X,M )

The Segal conditions (Section 12.2) for M -precategories can be ex-

pressed in terms of the algebraic diagram theory of the previous chapter,

which was the motivation for introducing that notion.

Let Φ := ∆oX , and let Φ0 = disc(X) be the discrete subcategory on

object set X , considered as a subcategory by letting x ∈ X correspond

to the sequence (x). An M -precategory A ∈ PC(X,M ) is by definition

the same thing as a functor Φ → M sending the objects of Φ0 to ∗,

which is to say

PC(X,M ) = Func(∆oX/X,M ).

The Segal conditions are a collection of finite product conditions as

was considered in the previous chapter. The set of product conditions Q

consists of the full set of objects of ∆oX . For q = (x0, . . . , xn) the integer

n(q) is equal to n, and we define a functor

P(x0,...,xn) : ǫ(n) → ∆oX

by

P(x0,...,xn)(ξ0) := (x0, . . . , xn),

P(x0,...,xn)(ξi) := (xi−1, xi) for 1 ≤ i ≤ n.

The images of the projection maps in ǫ(n) are the opposites of the in-

clusion maps (xi−1, xi) → (x0, . . . , xn) in ∆X . An M -precategory is

an M -enriched Segal category, if and only if it satisfies the product

condition with respect to the collection of functors P·, indeed the two

conditions are identically the same. Unitality gives the product condi-

tion whenever n = 0, whereas the product condition is automatically

true whenever n = 1 because the Segal maps are the identity in this

case. Thus, this condition needs only to be imposed for n ≥ 2.

We obtain adjoint functors

P(x0,...,xn),! : Func(ǫ(n),M ) ←→ Func(∆oX ,M ) : P ∗

(x0,...,xn)

and the unital versions

U!P(x0,...,xn),! : Func(ǫ(n),M ) ←→ PC(X,M ) : U!P∗(x0,...,xn)

where the right adjoint is just the pullback of a diagramA : ∆oX →M to

the category ǫ(n). An M -precategory A satisfies the Segal conditions, if

and only if U!P∗(x0,...,xn)

(A) is a product-compatible diagram ǫ(n) →M ,

for each sequence (x0, . . . , xn).

14.1 The model structures on PC(X,M ) 279

A direct application of the construction of Chapter 13 gives two model

structures (projective and injective) on PC(X,M ) such that the fibrant

objects satisfy the Segal condition. We add a third Reedy structure since

∆oX is a Reedy category.

The consideration of these model structures is intermediate with re-

spect to our main goal of constructing global model structures onPC(M ):

the maps in PC(X,M ) are ones which induce the identity on the set

of objects X . Nonetheless, these easier model structures on PC(X,M )

will be very useful in numerous arguments later.

The idea of introducing the intermediate model category PC(X,M ),

and of expressing it as a left Bousfield localization, is due to Barwick

[14].

Start by discussing the case of non-unital diagrams. If we fix gen-

erating sets (I, J) for M , then we obtain generating sets for the pro-

jective model structure Funcproj(∆oX ,M ) and injective model struc-

ture Funcinj(∆oX ,M ), recalled in Theorem 9.8.1. Notice that if M is

tractable and I and J consist of arrows with cofibrant domains, the ex-

plicit generators for the projective model structure also have cofibrant

domains. Thus Funcproj(∆oX ,M ) will again be tractable. For the in-

jective model structure the construction of generating sets of Barwick

and Lurie [16] [153] (as discussed in Theorem 8.9.2 above) was compli-

cated, and it doesn’t seem clear whether we can choose generators with

cofibrant domains. This problem can be bypassed later with the Reedy

model structures where again the generators become explicit.

Within the projective or injective model categories of non-unital di-

agrams, we obtain a direct left Bousfield localizing system (Rnu,Knu)

where R is the class of non-unital diagrams satisfying the Segal condi-

tions, and Knu is given by the generating trivial cofibrations plus the

maps of the form P(x0,...,xn),!(ζn(f)) for f in the generating set I of

cofibrations of M . Here ζn(f) is the diagram ǫ(n) →M considered in

Section 13.2.1. This yields the direct left Bousfield localized model struc-

tures which were designated by the notation Alg(. . .) in the previous

chapter. Denote these model categories of weakly unital precategories

now by

Algproj(∆oX , P·;M ), Alginj(∆

oX , P·;M ).

The underlying categories are both the same Func(∆oX ,M ). The fibrant

objects are diagrams which are fibrant in the projective or injective

model structures for diagrams, and which are Segal categories.

The same will work for for M -precategories where the unitality con-


dition is imposed. The model structure on M -precategories for a fixed

set of objects is given by the following theorem. Note the introduction of

the notation A → Seg(A) for a choice of fibrant replacement in either

of the model categories. This notation will be used later, but can mean

that a choice is made at each usage, rather than fixing a global choice

once and for all. Most constructions will be independent of the choice,

up to equivalence.

Theorem 14.1.1 Supose M is a tractable left proper cartesian model

category, then there are left proper combinatorial model category struc-

tures on the unital M -precategories

PCproj(X ;M ) := Algproj(∆oX/X, P·;M ),

PCinj(X ;M ) := Alginj(∆oX/X, P·;M ).

The fibrant objects are unital fibrant diagrams which satisfy the Segal

condition. The cofibrations are the projective or injective cofibrations in

the unital diagram category Func(∆oX/X ;M ). The weak equivalences

are the same in both structures.

Let A → Seg(A) denote a trivial cofibration towards a fibrant replace-

ment of A in the projective model structure PCproj(X ;M ); this can be

chosen functorially. A map A → B is a weak equivalence if and only if

Seg(A) → Seg(B) is a levelwise weak equivalence when considered as

a map of diagrams ∆oX → M .

Proof Propositions 13.4.3 and 13.4.2 give projective and injective di-

agram model structures Funcproj(∆oX/X,M ) and Funcinj(∆

oX/X,M )

on the category of unital diagrams, which is the same underlying cate-

gory as PC(X,M ).

We get direct left Bousfield localizing systems (R,Kproj) and (R,Kinj)

for these model structures, by transfering the direct localizing systems

for ǫ(n)-diagrams of Theorem 13.2.2, as was discussed in Section 13.5

using Theorem 11.8.3.

In both cases R is the class of M -precategories which satisfy the Segal

conditions; thenKproj (resp.Kinj) is the union of the set of generators for

trivial cofibrations in the projective (resp. injective) diagram structure,

plus the morphisms of the form U!P(x0,...,xn),!(ζn(f)) for f in the gener-

ating set I of cofibrations of M . Note that the images PU(x0,...,xn),!(g)

of generating trivial cofibrations for the diagram categories on ǫ(n(q))

are already trivial cofibrations in Func(∆oX/X,M ) so we don’t need to

include them again.

14.2 Unitalization adjunctions 281

Now Theorem 11.7.1 applies to give the required model structures. The

characterization of weak equivalences comes from Lemma 11.4.5.

14.2 Unitalization adjunctions

The projective model structure of Theorem 14.1.1 is related to the non-

unital version by a Quillen adjunction of unitalization

U! : Algproj(∆oX , P·;M ) ←→ PCproj(X ;M ) : U∗.

This follows from the application of Theorem 11.7.1.

It is useful to describe the unitalization operation for ∆oX/X .

Lemma 14.2.1 Suppose A : ∆oX → M is a functor. Then U!A has

the following explicit description:

—if x0 = . . . = xn = x is a constant sequence then the full degeneracy

gives a map A(x) → A(x, . . . , x) and

(U!A)(x, . . . , x) = A(x, . . . , x) ∪A(x) ∗;

—otherwise, if x0, . . . , xn is not a constant sequence then

(U!A)(x0, . . . , xn) = A(x0, . . . , xn).

Proof The object explicitly constructed in this way is again a functor

∆oX →M because if (x0, . . . , xn) is a constant sequence and (y0, . . . , yk) → (x0, . . . , xn)

is any map in ∆X then y· must also be a constant sequence, so the map

A(x0, . . . , xn) → A(y0, . . . , yk) passes to the quotients. The resulting

functor satisfies the required left adjunction property with respect to

U∗ on the right, so it must be U!A.

In the injective case, U! will not in general be a left Quillen functor;

we need to impose an additional hypothesis.

Condition 14.2.2 (INJ) Suppose

A → X → A

B

↓→ Y

↓→ B

↓

is a diagram in M such that the vertical arrows are cofibrations (resp.

trivial cofibrations) and the horizontal compositions are the identity.

Then X ∪A ∗ → Y ∪B ∗ is a cofibration (resp. trivial cofibration).


The following observations allow us to use the injective model cate-

gories in many cases. In fact, in these cases the injective structure also

coincides with the Reedy structure, however it seems comforting to be

able to use the injective structure which is conceptually simpler, instead.

Lemma 14.2.3 Suppose M is a presheaf category, is left proper, and

the class of cofibrations is the class of monomorphisms of presheaves.

Then Condition 14.2.2 holds.

Proof Note first that given a diagram of sets as in Condition 14.2.2

where the vertical maps are injections, then the map X ∪A ∗ → Y ∪B ∗

is an injection of sets. Indeed if x ∈ X maps to an element of B then

the image of x by the projection to A, maps to the same element of B.

By injectivity of the map X → Y it follows that x ∈ A, which shows

injectivity of the map on quotients.

Now if M is a presheaf category and the cofibrations are the monomor-

phisms, applying the previous paragraph levelwise we obtain the desired

result for cofibrations. Suppose given a diagram whose vertical arrows

are trivial cofibrations. The split injections A → X and B → Y are

monomorphisms, hence cofibrations, so Corollary 9.5.2 applies to con-

clude that the pushout map is a weak equivalence, hence it is a trivial

cofibration.

Lemma 14.2.4 If M satisfies Condition 14.2.2 then the unitalization

functors give a Quillen adjunction between injective diagram structures

U! : Alginj(∆oX , P·;M ) ←→ PCinj(X ;M ) : U∗

where U∗ is just the identity inclusion of unital precategories in all pre-

categories.

Proof We first show that unitalization is a Quillen adjunction between

levelwise injective diagram categories

U! : Funcinj(∆oX ,M ) ←→ Funcinj(∆

oX/X,M ) : U∗.

Suppose A → B is a levelwise cofibration (resp. trivial cofibration) of

diagrams, the claim is that U!A → U!B is a levelwise cofibration (resp.

trivial cofibration). In view of the description of Lemma 14.2.1 it suffices

to look at the values over a constant sequence (x, . . . , x). We have a


diagram

A(x) → A(x, . . . , x) → A(x)

B(x)

↓

→ B(x, . . . , x)

↓

→ B(x)

↓

where the vertical arrows are cofibrations (resp. trivial cofibrations), and

the second horizontal arrows are, say, the projections corresponding to

the first object of the sequence. Condition 14.2.2 now says exactly that

U!A(x, . . . , x) = A(x, . . . , x) ∪A(x) ∗

U!B(x, . . . , x) = B(x, . . . , x) ∪B(x) ∗

↓

is a cofibration (resp. trivial cofibration). This shows the Quillen adjunc-

tion for the diagram categories.

Now, the categories in question for the lemma are obtained by direct

left Bousfield localization using the sets of generators plus the morphisms

of the form Pq,!(ζn(q)(f)) for q ∈ Q and f in a generating set for cofi-

brations of M . The left Quillen functor passes to a left Quillen functor

between localizations by Theorem 11.8.1.

14.3 The Reedy structure

The category ∆oX is a Reedy category, using the subcategories of in-

jective and surjective maps of finite ordered sets, as direct and inverse

subcategories, and the length function (x0, . . . , xn) 7→ n. This leads to a

Reedy model structure on the category of diagrams, using levelwise weak

equivalences, denoted FuncReedy(∆oX ,M ) (Proposition 9.8.3).

The notion of Reedy structure on diagram categories is a slightly more

technical area of the theory of model categories, but these structures are

very natural and turn out to be the best ones for our theory of precate-

gories. In many useful examples the Reedy structure coincides with the

injective structure. This is the case for example if M = Presh(Ψ) is a

presheaf category and the cofibrations of M are the monomorphisms of

presheaves, see Proposition 15.7.2.


There is a unital version of the Reedy model structure.

Proposition 14.3.1 For any tractable left proper model category M ,

the unital diagram category has a tractable left proper model structure

denoted FuncReedy(∆oX/X,M ), related to the non-unital Reedy struc-

ture by a Quillen adjunction

U! : FuncReedy(∆oX ,M ) ←→ FuncReedy(∆

oX/X,M ) : U∗

using the levelwise weak equivalences. The fibrations (resp. cofibrations)

of FuncReedy(∆oX/X,M ) are exactly the maps f such that U∗(f) is a

Reedy fibration (resp. cofibration) in FuncReedy(∆oX ,M ), in particular

they have the same description in terms of latching and matching ob-

jects. The generating sets for the unital Reedy structure are obtained by

applying U! to the generating sets for the regular Reedy diagram struc-

ture.

The Reedy structure lies in between the projective and injective struc-

tures with a diagram of left Quillen functors

Funcproj(∆oX ,M ) → FuncReedy(∆

oX ,M ) → Funcinj(∆

oX ,M )

Funcproj(∆oX/X,M )

U!

↓

→ FuncReedy(∆oX/X,M )

U!

↓

→ Funcinj(∆oX/X,M )

where the horizontal rows are identity functors. If M satisfies Condition

14.2.2 then this can be completed by putting in the rightmost vertical

arrow.

Proof The first thing to show is that if Af→ B is a Reedy cofibration of

diagrams on ∆oX , then U∗U!A

U∗U!f→ U∗U!B is again a Reedy cofibration.

The latching objects over a sequence (x0, . . . , xn) ∈ ∆oX involve maps

which correspond to surjections (x0, . . . , xn)σ→ (y0, . . . , yk) of sequences,

with σ : [n] ։ [k] with yσ(i) = xi. In particular, any (y0, . . . , yk) involved

in the latching map at a non-constant sequence (x0, . . . , xn), is also not

constant. Over these sequences the relative latching map for U∗U!f is

the same as for f so the Reedy condition is preserved. We may therefore

concentrate on the case of a constant sequence. Let xn := (x0, . . . , xn)

with xi = x. Let A′ := U∗U!A, B′ := U∗U!B, and f ′ := U∗U!f . Thus

A′(xn) = A(xn) ∪A(x) ∗, B′(xn) = B(xn) ∪B(x) ∗.


The latching object for A is expressed as the pushout∐

0<i<j≤n

A(xn−2) →∐

0<i≤n

A(xn−1)

∐

0<i≤n

A(xn−1)

↓

→ latch(A, xn)

↓

and similarly for latch(B, xn). The coproducts may be considered as be-

ing taken over contractions of adjacent pairs in the sequence (x0, . . . , xn)

with all xi = x.

We get a map from A(x) into all elements of the above cocartesian

diagram, and then the same diagram for A′ gives

latch(A′, xn) = latch(A, xn) ∪A(x) ∗.

Similarly

latch(B′, xn) = latch(B, xn) ∪B(x) ∗.

Recall that

latch(f, xn) := latch(B, xn) ∪latch(A,xn) A(xn) → B(xn).

Thus latch(f ′, xn) is obtained by contracting out B(x) in the domain

and range, in particular

latch(B′, xn)∪latch(A′,xn)A′(xn) = ∗∪B(x) latch(B, xn)∪latch(A,x

n)A(xn)

as can be seen by using the universal property of coproducts. In the

diagram

B(x) → latch(B, xn) ∪latch(A,xn) A(xn) → B(xn)

∗↓

→ latch(B′, xn) ∪latch(A′,xn) A′(xn)

↓

→ B′(xn)

↓

the left square is cocartesian, and the outer square is cocartesian, there-

fore the right square is cocartesian. The upper right map is assumed to

be a cofibration, so the bottom right map is also a cofibration.

This shows that U∗U!f is a Reedy cofibration whenever f is. The same

argument shows that if f is a Reedy trivial cofibration then U∗U!f is a

Reedy trivial cofibration.


To construct the model structure on the unital diagram category, de-

fine the weak equivalences (resp. cofibrations, fibrations) to be those

arrows f in FuncReedy(∆oX/X,M ) such that U∗f is a weak equivalence

(resp. cofibration, fibration). In view of this definition, the same holds

for the intersection classes of trivial fibrations and trivial cofibrations.

Since U∗ takes compositions to compositions and retracts to retracts,

the classes are all closed under retracts, and weak equivalences (which

are levelwise weak equivalences) satisfy 3 for 2.

Suppose IR and JR are generating sets for the Reedy structure on

non-unital diagrams FuncReedy(∆oX ,M ), and let I ′R := U!IR and J ′

R :=

U!JR. Apply Proposition 9.2.1. Note that elements of I ′R are cofibrations

(CG2a) and elements of J ′R are trivial cofibrations (CG3a), by the above

arguments. All objects are small (CG1).

To complete, we need to show that a map g : U → V is a fibration

(resp. trivial fibration) if and only if it satisfies lifting with respect to

I ′R (resp. J ′R). But this is an immediate consequence of the definitions

and the adjunction property. For example, g is a fibration ⇔ U∗g is a

fibration ⇔ U∗g satisfies right lifting with respect to JR ⇔ g satisfies

right lifting with respect to U!JR = J ′R. The proof for trivial fibrations

is the same.

Applying Proposition 9.2.1 we get a cofibrantly generated model cat-

egory, from which it also follows that cof(I ′R) is the class of cofibrations,

and cof(J ′R) is the class of trivial cofibrations.

The Reedy diagram structure is tractable, so we may assume that

the domains of elements of IR and JR are cofibrant, from which it fol-

lows that the same is true of I ′R and J ′R. Reedy cofibrations are in-

jective cofibrations. But Reedy-unital cofibrations (i.e. cofibrations in

FuncReedy(∆oX/X,M )) are maps whose U∗ are Reedy diagram cofibra-

tions, hence they are levelwise cofibrations. Also weak equivalences are

defined levelwise; so left properness may be verified levelwise.

For the second paragraph of the theorem, the upper sequence of left

Quillen functors

Funcproj(∆oX ,M ) → FuncReedy(∆


oX ,M )

is classical. From this the lower sequence follows too. To go from the

projective to the Reedy structure, the generators of the projective uni-

tal diagram structure are obtained by applying U! to generators of the

projective non-unital structure, but these go to Reedy cofibrations (resp.

trivial cofibrations) in the non-unital Reedy structure, which in turn get

sent to the same in the unital Reedy structure. To go from the Reedy to


the injective structure, note that the unital Reedy cofibrations are also

non-unital Reedy cofibrations by definition, and these are levelwise cofi-

brations by the upper sequence. The same is true for trivial cofibrations

because the weak equivalences are the same in all cases.

If M satisfies Condition 14.2.2 then U! is a left Quillen functor by

Lemma 14.2.4 and it clearly makes the diagram commute. This com-

pletes the proof.

Theorem 14.3.2 Suppose M is a tractable left proper cartesian model

category. The model structure of FuncReedy(∆oX/X,M ) admits a direct

left Bousfield localization denoted PCReedy(X,M ) by a system (R,KReedy)

where R is the class M -precategories satisfying the Segal conditions, and

KReedy is the generating set for trivial Reedy cofibrations, plus the set

of cofibrations of the form U!P(x0,...,xn),!(ρn(f)) for f a generating cofi-

bration of M . The cofibrations of PCReedy(X,M ) are the maps which

are Reedy cofibrations in the diagram category FuncReedy(∆oX ,M ). The

fibrant objects of PCReedy(X,M ) are the Reedy-fibrant diagrams which

satisfy the Segal conditions. The Reedy structure lies between the projec-

tive and injective structures with the identity functors being left Quillen

PCproj(X,M ) → PCReedy(X,M ) → PCinj(X,M ).

The weak equivalences are the same for the three structures.

Proof This is analogous to the proof of Theorem 14.1.1 using Theorem

11.7.1 and Theorem 11.8.3. The only difference worth pointing out here

is that we can use the morphisms ρn(f) defined just before Proposition

13.1.4, in place of the ζn(f) from Section 13.2.1.

We claim that

U!P(x0,...,xn),!(n(f))U!P(x0,...,xn),!(ρn(f))

→ U!P(x0,...,xn),!(ξ0,!(V ))

is a Reedy cofibration if Uf→ V is a cofibration in M . Recall from

Equation (13.1.1) that

n(f) := ξ0,!(U) ∪∐

i ξi,!(U)∐

i

ξi,!(V ) = (U, V, . . . , V ; f, . . . , f).

It will be more convenient to consider an arbitrary A ∈ PC(X,M ) with

a map U!P(x0,...,xn),!(n(f))a→ A and let

B := A∪U!P(x0,...,xn),!(n(f)) U!P(x0,...,xn),!(ξ0,!(V ))

be the pushout along U!P(x0,...,xn),!(ρn(f)). We would like to show that

A → B is a cofibration.


The map a corresponds to a collection of maps ai : V → A(xi−1, xi)

and a0 : U → A(x0, . . . , xn) such that the diagrams

U → A(x0, . . . , xn)

V

↓→ A(xi−1, xi)

↓

commute. We can now describe B explicitly. For any sequence (y0, . . . , yk),

B(y0, . . . , yk) = A(y0, . . . , yk) ∪∐

y·→x·U

(∐

y·→x·

V

),

where the coproduct is taken over all arrows y· → x· which don’t factor

through one of the adjacent pairs (xi−1, xi) → (x0, . . . , xn). The left

map in the coproduct uses a0, whereas the functoriality maps to define

the functor B : ∆oX →M are made using the ai for i = 1, . . . , n.

Suppose (z0, . . . , zl) → (y0, . . . , yk) is a map. For any h : y· → x· which

doesn’t factor through an adjacent pair, if the composition z· → x·does factor through an adjacent pair (xi−1, xi) then we use the map ai :

V → A(xi−1, xi)→ A(z0, . . . , xl) on the component V of B(y0, . . . , yk)

corresponding to h. The other maps of functoriality are straightforward.

From this description we already see easily that A → B is an injective

i.e. levelwise cofibration. To show that it is a Reedy cofibration, con-

sider (y0, . . . , yk). The latching objects are colimits over the category of

surjections (y0, . . . , yk)σ։ (z0, . . . , zl) with l < k. For any σ, the arrows

z· → x· which don’t factor through an adjacent pair, form a subset of

the similar arrows y· → x·. We can write

latch(B, y·) = latch(A, y·) ∪∐

y·→x·,factU

∐

y·→x·,fact

V

where the coproduct is now taken over y· → x· which factor through

some surjection y· → z· in the latching category. The relative latching

object for the map A → B is then

A(y·) ∪∐

y·→x·,factU

∐

y·→x·,fact

V

→ A(y·) ∪

∐y·→x·

U

(∐

y·→x·

V

).

Thus, the relative latching map consists just of adding on some addi-

14.4 Some remarks 289

tional pushouts along U → V . It is cofibrant, which shows that A → B

is a Reedy cofibration.

Now the same proof as for Theorem 14.1.1 shows that we can get

a direct localizing system for the class R of M -enriched Segal cat-

egories by starting with the generating set for trivial cofibrations in

FuncReedy(∆oX/X,M ) and adding the U!P(x0,...,xn),!(ρn(f)) to getKReedy.

We could also have added the U!P(x0,...,xn),!(ζn(f)), or both. These

all give the same left Bousfield localization (Proposition 11.7.2). Using

the U!P(x0,...,xn),!(ζn(f)), Theorem 11.8.1 gives the left Quillen functors

from the projective to the Reedy and then to the injective structure.

We can remark that the above proof also shows that we could have

used the U!P(x0,...,xn),!(ρn(f)) to form a direct localizing system for the

injective structure.

14.4 Some remarks

The model structures on PC(X ;M ) are the main intermediate struc-

ture to be considered [14] before getting the model structures for moving

object sets. In the remainder of the book, when we speak of weak equiv-

alences or the injective or projective model structures on PC(X ;M ),

we mean the structures of Theorem 14.1.1. In case it is necessary, we use

the terminology levelwise weak equivalence to speak of weak equivalences

in the unital diagram category Func(∆oX/X,M ).

Lemma 14.4.1 If A,B ∈ PC(X ;M ) satisfy the Segal conditions, then

a map f : A → B in PC(X ;M ) is a weak equivalence if and only if it

is a levelwise weak equivalence.

If B satisfies the Segal conditions and A → B is a trivial fibration in

either the projective or injective model structures on PC(X ;M ), then A

also satisfies the Segal conditions and f is a levelwise weak equivalence.

Proof Consider the square

A → Seg(A)

B

f

↓→ Seg(B).

Seg(f)

↓

If A and B both satisfy the Segal conditions, then the horizontal arrows

are levelwise weak equivalences. By the definition of the model structure


of Theorem 14.1.1 the map f is a weak equivalence if and only if Seg(f)

is a levelwise weak equivalence, and by 3 for 2 this is equivalent to

requiring that f be a levelwise weak equivalence.

Suppose B satisfies the Segal conditions, and f : A → B is a trivial

fibration in the projective model structure on PC(X,M ). This means

that f satisfies right lifting with respect to cofibrations, but the cofibra-

tions are the same as those of the unital diagram category. Thus, f is a

trivial fibration in Func(∆oX/X,M ), in particular it is a levelwise weak

equivalence. Under our hypotheses on M , Lemma 10.0.11 says that di-

rect product is invariant under weak equivalences in M , from which it

follows that A satisfies the Segal conditions too.

In the diagram category Func(∆oX ,M ) limits and colimits are com-

puted levelwise: if Aii∈α is a diagram of objects Ai ∈ Func(∆oX ,M )

then

(colimi∈αAi)(x0, . . . , xn) = colimi∈α(Ai(x0, . . . , xn)),

and

(limi∈αAi)(x0, . . . , xn) = lim

i∈α(Ai(x0, . . . , xn)).

The right adjoint functor U∗ being the identity means that the same

holds true for all limits in PC(X ;M ): if Aii∈α is a diagram of objects

Ai ∈ PC(X ;M ) then

(limi∈αAi)(x0, . . . , xn) = lim

i∈α(Ai(x0, . . . , xn)).

On the other hand, for colimits in PC(X ;M ) we have to reapply the

functor U!. Write this with superscriptsPC or Func to indicate in which

category the colimit is taken:

colimPCi∈αAi = U!(colim

Funci∈α Ai).

In the special case where colimFunci∈α Ai is already in PC(X,M ) then

the functor U! acts as the identity. The condition that colimFunci∈α Ai

already be in PC(X,M ), says that the value on sequences of length

zero (x0) is ∗. As the colimFunc is calculated levelwise, it is sufficient

to require that colimα∗ = ∗ which holds when α is a connected category

by Lemma 8.1.8. This condition is necessary and sufficient unless X is

the empty set.

Lemma 14.4.2 Suppose α is a connected category and Aii∈α is a


diagram of objects Ai ∈ PC(X ;M ). Then the colimit in PC(X ;M )

may be calculated levelwise:

(colimPCi∈αAi)(x0, . . . , xn) = colimi∈α(Ai(x0, . . . , xn)).

Proof Above (also, this is the same as the corresponding statement in

Theorem 13.4.1 of the previous chapter).

Corollary 14.4.3 Suppose M is locally κ-presentable, and X is a set

with < κ elements. An object A ∈ PC(X ;M ) is κ-presentable if and

only if each A(x0, . . . , xp) is a κ-presentable object of M . The category

PC(X ;M ) is locally κ-presentable.

Proof Apply Theorem 13.4.1.

For colimits over disconnected categories, in particular for disjoint

sums, one has to apply the functor U!. This will not be used very often,

because when we work in the category PC(M ) with variable set of

objects which will be defined next, disjoint sums lead to disjoint unions

of sets of objects, returning a reasonable description (Section 12.6).

14.5 Global weak equivalences

We turn now from consideration of the various model categories for M -

precategories with a fixed set of objects, to the global category PC(M )

of M -precategories with arbitrary variable set of objects. We’ll get back

to a closer analysis of the generation operation Seg on PC(X,M ) in

Chapter 16 on the calculus of generators and relations, but first we

consider weak equivalences in the global case, and in the next Chapter

15, various types of cofibrations in the global case.

In this section we formalize Tamsamani’s induction step for defin-

ing equivalences of n-categories, as applied in the M -enriched case.

For classical Segal categories which is the K -enriched case, this notion

of equivalence goes back at least to Dwyer and Kan and is known as

“Dwyer-Kan equivalence” [178] [38]. The general case requires a functor

τ≤0 : M → Set in order to be able to talk about the essential surjectiv-

ity condition; this issue was explored by Pelissier, but we have modified

his situation a little bit by introducing sets X of objects external to M .

Our discussion here specializes the more general discussion in Section

3.3.


Define the 0-th truncation functor τ≤0 : M → Set by

τ≤0(A) := Homho(M )(∗,A).

Then define a truncation functor τ≤1 : PC(M ) → Cat as follows: for

A ∈M with Ob(A) = X , choose a weak equivalence in PC(X,M ) from

A to an M -enriched Segal category Seg(A) (i.e. a precategory which

satisfies the Segal conditions). Consider the functor from ∆X to Set

defined by

τ≤1(A)(x0, . . . , xn) := τ≤0(Seg(A)(x0, . . . , xn))

= Homho(M )(∗,Seg(A)(x0, . . . , xn)).

The Segal maps for this functor are isomorphisms (proven in the next

lemma), so it defines a 1-category whose object set isX and such that the

set of morphisms from x0 to x1 is τ≤1(A)(x0, x1) = τ≤0(Seg(A)(x0, x1).

One good choice for Seg(A) would be to take a fibrant replacement,

however other smaller choices can be useful too (see Chapter 16 below).

Lemma 14.5.1 The truncation τ≤0 : M → Set sends weak equiva-

lences to isomorphisms and products to products. The Set-precategory

τ≤1(A) defined above satisfies the Segal conditions so it corresponds to

a 1-category, and this category is independent of the choice of Seg(A).

It gives a functor τ≤1 : PC(M ) → Cat, and for a fixed object set X

the functor takes weak equivalences in PC(X,M ) to isomorphisms.

Proof The τ≤0 factors through the projection to the homotopy category

so it clearly preserves weak equivalences. It preserves products since M

is assumed to be cartesian. In particular the Segal condition for Seg(A)

yields the Segal condition for the ∆oX -set τ≤1(A) to be the nerve of a

category, independent of the choice of Seg(A) because τ≤0 sends weak

equivalences to isomorphisms. We get a functor τ≤1. If A → B is a weak

equivalence in PC(X,M ) then Seg(A) → Seg(B) is a levelwise weak

equivalence (see Theorem 14.1.1) so the resulting τ≤1(A) → τ≤1(B) is

fully faithful, but in the present case it is also the identity on the set X

of objects so it is an isomorphism.

Lemma 14.5.2 If A and B are M -precategories satisfying the Segal

condition then there is a natural isomorphism

τ≤1(A× B) ∼= τ≤1(A)× τ≤1(B).

Proof As τ≤0 is compatible with products this follows from the defini-

tion of τ≤1.


The truncation operation is crucial to Tamsamani’s construction of

n-categories [206], since it allows us to define equivalences between en-

riched category objects. In the case of n-categories, M corresponds to

the model category for n−1-categories and ho(M ) can be identified with

the homotopy category of Segal n− 1-categories. For Segal n-categories,

the truncation as we have defined it here coincides with Tamsamani’s

truncation operation which was constructed by induction on n. The gen-

eral iteration procedure requires a general definition of truncation start-

ing from a model category M . This more general case was considered

by Pelissier [171, Definition 1.4.1].

We now proceed with the definition of global weak equivalence, com-

bining two conditions: “fully faithful” is defined by asking for weak equiv-

alences in M between morphism objects, and “essentially surjective” is

defined using the above truncation operation.

A map f : (X,A) → (Y,B) consists of a map Ob(f) : X → Y

which we sometimes denote by f for short, togegher with maps f :

A(x0, . . . , xp) → B(f(x0), . . . , f(xp)) in M for each (x0, . . . , xp) ∈ ∆X ,

compatible with restrictions along maps in ∆X .

Suppose that (X,A) and (Y,B) are M -precategories satisfying the

Segal condition. A map f : (X,A) → (Y,B) is fully faithful if, for every

sequence (x0, . . . , xn) of objects in X the map

f(x0, . . . , xn) : A(x0, . . . , xn) → B(f(x0), . . . , f(xn))

is a weak equivalence in M .

If (X,A) and (Y,B) are only M -precategories, let A → Seg(A) and

B → Seg(B) denote injective trivial cofibrations towards objects satisfy-

ing the Segal conditions—for example, fibrant replacements in either the

injective model structures PCinj(X,M ) and PCinj(Y,M ). Recall that

Ob(f)∗(B) is an M -precategory on X = Ob(A), as is Ob(f)∗(Seg(B)).

Furthermore, the map Ob(f)∗(B) → Ob(f)∗(Seg(B)) is again a weak

equivalence in the model structure PCinj(X,M ) (see Lemma 12.3.2). It

is also again a levelwise cofibration, so it is again a trivial cofibration.

By making an appropriate choice of construction of the operation

Seg(·) we can obtain that f induces a map Seg(A) → Ob(f)∗(Seg(B));

however, in any case we could modify the choice of Seg(A) in order to

obtain such a factorization. So we shall assume that this factorization is

given. This results in a map Seg(A) → Seg(B).

We say that f : A → B is fully faithful if the map Seg(A) → Seg(B) is


fully faithful in the previous sense, i.e. it induces a levelwise equivalence

Seg(A)(x0, . . . , xp)∼→ Seg(B)(f(x0), . . . , f(xp))

for each sequence of objects (x0, . . . , xp) ∈ ∆XOb(A). Since both Seg(A)

and Seg(B) satisfy the Segal condition, it is easy to see that it is suffi-

cient to check this on sequences of length one, that is it suffices to require

that

Seg(A)(x, y)∼→ Seg(B)(f(x), f(y))

for every pair of objects x, y ∈ Ob(A).

We say that a map f : (X,A) → (Y,B) is essentially surjective if

the functor of categories obtained by the truncation operation τ≤1(f) :

τ≤1(X,A) → τ≤1(Y,B) is an essentially surjective map of categories,

in other words if the set of isomorphism classes Isoτ≤1(A) of τ≤1(X,A)

surjects to the set of isomorphism classes Isoτ≤1(B) of τ≤1(Y,B).

We say that f : (X,A) → (Y,B) is a global weak equivalence if it is

fully faithful and essentially surjective.

Lemma 14.5.3 A morphism f : A → B which induces an isomor-

phism on sets of objects Ob(f) : Ob(A) ∼= Ob(B) is a global weak

equivalence if and only if the corresponding map f♯ : Ob(f)!A → B

in PC(Ob(B),M ) is a weak equivalence in the model structures of The-

orems 14.1.1 and 14.3.2.

In particular, if Ob(f) is an isomorphism and f♯ is a levelwise weak

equivalence of ∆oOb(B)-diagrams in M , then f is a weak equivalence.

Proof This is immediate from the definitions, since the essential sur-

jectivity is automatically guaranteed by the condition that Ob(f) be an

isomorphism. For the last paragraph note that Corollary 11.3.4 applies in

the direct left Bousfield localizations of Theorems 14.1.1 and 14.3.2.

14.6 Categories enriched over ho(M )

As in the proof of the preceding propsition, using the small object argu-

ment for the pseudo-generating sets for trivial cofibrations inPCproj(X,M ),

we can obtain a functorial construction which associates to any A ∈

PC(M ) another Seg(A) ∈ PC(Ob(A),M ) with a natural transforma-

tion A → Seg(A) which is a weak equivalence in PC(Ob(A),M ) such

that Seg(A) satisfies the Segal condition. For the present section, fix one

14.6 Categories enriched over ho(M ) 295

such construction, although the discussion is left invariant if we make a

different choice.

The category ho(M ) admits finite direct products, and these are cal-

culated by direct products in M using conditions (DCL) and (PROD)

(Lemma 10.0.11). Therefore it makes sense to look at the category

Cat(ho(M )) of ho(M )-enriched categories, see Section 8.5. Recall that

a ho(M )-enriched category C consists of a set Ob(C) and for any x, y ∈

Ob(C), a morphism object C(x, y) ∈ ho(M ), plus composition maps

satisfying strict associativity.

Lemma 14.6.1 Suppose A ∈ PC(M ). Then we obtain a ho(M )-

enriched category denoted eh(A) with the same set of objects as A,

by putting eh(A)(x, y) equal to the class of Seg(A)(x, y) in ho(M ).

The remainder of the simplicial diagram structure of Seg(A) (up to

Seg(A)(x0, x1, x2, x3)) provides this with composition maps which are

unital and strictly associative. This gives a functor from PC(M ) to

Cat(ho(M )).

Proof Composing ∆oX

Seg(A)→M with the projection M → ho(M )

we get a functor eh(A) : ∆oX → ho(M ). The homotopy Segal conditions

for Seg(A) imply that the Segal maps for eh(A) are isomorphisms. By

the interpretation of Theorem 8.5.4 (the reader is encouraged by now to

have looked at the proof of this theorem, which was left as an exercise),

this says that eh(A) corresponds to a ho(M )-enriched category.

Recall that the truncation τ≤0 on M factors through a functor denoted

the same way τ≤0 : ho(M ) → Set compatible with direct products, so

and applying that to the enrichment category gives a functor

τh≤1 : Cat(ho(M )) → Cat.

The image of eh(A) under this functor is exactly τ≤1(A) according to

its construction, that is to say

τh≤1(eh(A)) = τ≤1(A). (14.6.1)

Recall from Section 8.5 that a functor g : C → C′ in Cat(ho(M )) is an

equivalence of categories if g essentially surjective i.e.

Isoτh≤1(g) : Isoτh≤1(C) → Isoτh≤1(C

′)

is surjective, and if g is fully faithful i.e. for any x, y ∈ Ob(C) the map

C(x, y) → C′(g(x), g(y)) is an isomorphism in ho(M ). As in Lemma


8.5.1, if g is an equivalence of categories then Isoτh≤1(g) is an isomor-

phism.

Proposition 14.6.2 A morphism of M -precategories f : A → B is

a global weak equivalence in PC(M ) if and only if the corresponding

morphism eh(f) : eh(A) → eh(B) is an equivalence of ho(M )-enriched

categories.

Proof By the identification (14.6.1) between truncations, the essential

surjectivity condition is the same. The fully faithful conditions are the

same too, because a morphism in M is an equivalence if and only if its

image in ho(M ) is an isomorphism.

This point of view allows us to apply the arguments of Section 8.5,

which concerned a simpler 1-categorical notion of enrichment, to obtain

some important first properties of the class of global weak equivalences.

Lemma 14.6.3 Suppose f is a global weak equivalence. Then τ≤1(f) :

τ≤1(A) → τ≤1(B) is an equivalence of categories, in particular it induces

an isomorphism of sets Isoτ≤1(A) ∼= Isoτ≤1(B).

Proof Use Lemma 8.5.1.

Proposition 14.6.4 The class of global weak equivalences is closed

under retracts and satisfies 3 for 2. Furthermore if f : A → B and g :

B → A are morphisms such that fg and fg are global weak equivalences,

then f and g are global weak equivalences.

Proof Using Proposition 14.6.2, this becomes an immediate consequence

of Theorem 8.5.3.

14.7 Change of enrichment category

The following discussion is the key to resolving the issue that was raised

by Pelissier in [171]: he found a mistake in [193], in what will correspond

to our construction of interval objects in Chapter 20 below. Pelissier

solved the problem for M = K , but one can go pretty easily from

there to any M by investigating what happens under change of enrich-

ment category. This strategy will be used to construct interval objects

in Chapter 20.

Suppose F : M →M ′ is a left Quillen functor between tractable left

proper cartesian model categories; denote its right adjoint by F ∗. Then


F it induces left Quillen functors between the levelwise diagram model

categories

F∆oX : Funcinj(∆


oX ,M

′)

and

F∆oX : Funcproj(∆

oX ,M ) → Funcproj(∆

oX ,M

′)

whose right adjoints are obtained by applying the right adjoint F ∗ level-

wise. The Quillen property is verified levelwise on the left for the injective

structure and levelwise on the right for the projective structure.

Define the functor

PC(X,F ) : PC(X,M ) → PC(X,M ′)

by putting

PC(X,F )(A) := U!(F∆o

X (U∗A)).

The reader may note that if we assume furthermore F (∗) = ∗ then

PC(X,F ) is just the restriction of F∆oX to the unital diagrams, and the

following discussion could be simplified. We don’t make that assumption

in general.

Lemma 14.7.1 The functor PC(X,F ) is left adjoint to the functor

PC(X,F ∗) obtained by restricting the levelwise adjoint to the unital

diagram categories. It is compatible with F∆oX via the diagram

Func(∆oX ,M )

F∆oX

→ Func(∆oX ,M

′)

PC(X,M )

U!

↓PC(X,F )

→ PC(X,M ′).

U!

↓

Proof Note that F ∗ commutes with limits so F ∗(∗) = ∗, in particu-

lar applying F ∗ levelwise preserves the unitality condition, defining a

functor PC(X,M ′)PC(X,F∗)

→ PC(X,M ). We verify that PC(X,F )

as defined above is its left adjoint. Suppose A ∈ PC(X,M ) and B ∈

PC(X,M ′). A map PC(X,F )(A) → B in PC(X,M ′) is the same

as a map F∆oX (U∗A) → U∗B which in turn is the same as a map

U∗A → F ∗,∆oXU∗B, but this is the same as a map A → PC(X,F ∗)B.

For the compatibility diagram note that

PC(X,F )(U!A) = U!(F∆o

X (U∗U!A)) = U!(F∆o

X (A)).


This can be seen by the explicit description of the unitalization operation

U∗U! as a coproduct at the constant sequences

U!(F∆o

X (U∗U!A))(xn) = F (U∗U!A(x

n)) ∪F (U∗U!A(x)) ∗

= F (A(xn) ∪A(x) ∗) ∪F (∗) ∗ = F (A(xn)) ∪F (A(x)) ∗ = U!(F∆o

X (A))(xn).

Recall that the unitalization is trivial at nonconstant sequences.

Assume that M and M ′ are tractable left proper cartesian model

categories. If F : M →M ′ is a left Quillen functor, then for any finite

collection of objects A1, . . . , Am (including the empty collection with

m = 0) we have

F (A1 × · · · ×Am)→ F (A1)× · · · × F (Am).

We say that F is product-compatible if these maps are weak equivalences

for any sequence of objects. In particular for the case m = 0 this says

that F (∗) is contractible.

Strangely enough, this natural condition does not seem to be needed

for the following proposition.

Proposition 14.7.2 Suppose that MF→M ′ is a left Quillen func-

tor between two tractable left proper cartesian model categories. Then

the functor PC(X ;F ) is a left Quillen functor between the projective

(resp. Reedy) model category structures on PC(X,M ) and PC(X,M ′).

If M ′ furthermore satisfies Condition 14.2.2 then PC(X ;F ) is also a

left Quillen functor between the injective structures.

Proof We first note that PC(X,F ) is a left Quillen functor between

unital diagram categories

Func(∆oX/X,M ) → Func(∆o

X/X,M′)

in the projective and Reedy structures (and the injective structures if

Condition 14.2.2 holds).

For the projective structure, the fibrations and trivial fibrations are

defined levelwise, but the right adjoint PC(X,F ∗) is also defined level-

wise so it preserves levelwise (trivial) fibrations. For the Reedy structure,

recall that the diagrammatic Reedy structure is defined using the inclu-

sion U∗ from unital to all diagrams: a morphism f of unital diagrams

is a Reedy (trivial) cofibration if and only if U∗f is. So F∆oX (U∗ sends

Reedy (trivial) cofibrations in FuncReedy(∆oX/X,M ) to Reedy (triv-

ial) cofibrations in FuncReedy(∆oX ,M ). But now U! is left Quillen for


the Reedy structure by Proposition 14.3.1, so PC(X,F ) = U!F∆o

X (U∗

preserves Reedy (trivial) cofibrations.

If we assume furthermore Condition 14.2.2 then by Lemma 14.2.4, U!

is left Quillen for the injective structures so the same argument works.

This completes the proof of what is claimed in the first paragraph.

Now PC(X ;F ) passes to a left Quillen functor between the direct

left Bousfield localizations. We may assume that F sends the generat-

ing sets for M into ones for M ′. Then the functor PC(X ;F ) sends the

pseudo-generators P(x0,...,xn),!(ζn(f)) for the direct left Bousfield local-

ized structure on PC(X,M ), to pseudo-generators for PC(X,M ′); in

particular to trivial cofibrations. By the left Bousfield localization prop-

erty PC(X ;F ) are left Quillen functors between the direct left Bousfield

localizations.

15

Cofibrations

In this chapter, continuing the construction of model structures for the

category PC(M ) of M -enriched precategories with variable set of ob-

jects, we define and discuss various classes of cofibrations. The corre-

sponding classes of trivial cofibrations are then defined as the intersec-

tion of the cofibrations with the weak equivalences defined in the pre-

ceding chapter. The fibrations are defined by the lifting property with

respect to trivial cofibrations. It will have to be proven later that the

class of trivial fibrations, defined as the intersection of the fibrations

and the weak equivalences, is also defined by the lifting property with

respect to cofibrations.

In reality, we consider three model structures, called the injective, the

projective, and the Reedy structures. These will be denoted by subscripts

I, P,R respectively when necessary. They share the same class of weak

equivalences, but the cofibrations are different. It turns out that the

Reedy structure is the best one for the purposes of iterating the con-

struction. That is a somewhat subtle point because the Reedy structure

coincides with the injective structure when M lies in a wide range of

model categories where the monomorphisms are cofibrations, see Propo-

sition 15.7.2. When they are different it is better to take the Reedy

route.

15.1 Skeleta and coskeleta

The definition of Reedy cofibrations, as well as the study of the pro-

jective ones, is based on consideration of the skeleta and coskeleta of

objects in PC(M ). The Reedy structure will be defined in an explicit

way without making use of the general definition of Reedy category. The



category PC(M ) consists of precategories with various different object

sets X , so it isn’t enough to just invoke the Reedy structure of each ∆X .

Considering instead the relevant structures explicitly has the added ad-

vantage of making the discussion accessible for readers who wish to avoid

plunging into the full theory of Reedy categories. Those already famil-

iar with those kinds of things can try to view the discussion in a more

general language (see Barwick [17], Berger-Moerdijk [32] and others).

For a fixed set X , consider the subcategory ∆X,m ⊂ ∆X consisting of

all sequences (x0, . . . , xp) of length p ≤ m. For example, ∆X,0 consists of

sequences of length 0 i.e. (x0) only. Let σ(X,m) : ∆X,m → ∆X denote

the inclusion functor. Then we have a functor

skm := σ(X,m)! σ(X,m)∗ : Func(∆X ,M ) → Func(∆X ,M ).

We call skm(A) = σ(X,m)!(σ(X,m)∗(A)) the m-skeleton of A. There is

a natural map skm(A) → A.

Given m and A we would like to calculate skm(A). Suppose x· =

(x0, . . . , xk) is a sequence of elements of X of length k. Consider the

category of all surjections x· ։ y· where y· = (y0, . . . , yp) is a sequence of

length p ≤ m. Morphisms are morphisms of objects y· in ∆X commuting

with the maps from x·. Note that y· ∈ ∆X,m so the pullback of A

to ∆X,m restricts, by the projection functor to the variable y·, to a

contravariant functor from our category of surjections to M . We claim

that

skm(A)(x·) = colimx·։y·A(y·). (15.1.1)

Indeed, this is almost exactly the same as the expression of Lemma 8.4.1

for the pushforward σ(X,m)!. In the general expression, the colimit is

taken over all morphisms from x· to some y· ∈ ∆X,m. However, any

such morphism canonically factors as x· ։ y′· → y· with y′· ∈ ∆X,m too,

so the category of surjections is cofinal and suffices for calculating the

colimit.

From this expression we conclude that if x· = (x0, . . . , xk) is a se-

quence of length k ≤ m then the natural map

skm(A)(x·) → A(x·)

is an isomorphism, in other words the restrictions of A and skm(A)

to ∆X,m are the same. It follows that skm is a monad, in other words

skm(skm(A)) = skm(A) (or rather they are isomorphic in a natural way).

We say that A is m-skeletic if the map skm(A) → A is an isomorphism;

302 Cofibrations

and skm is a monadic projection to the full subcategory of m-skeletic

objects.

If n ≥ m andA ism-skeletic then it is also n-skeletic, as skn(skm(A)) =

skm(A). This formula comes from the fact that the restriction to ∆X,n of

skm(A) is the same as the pushforward from ∆X,m to ∆X,n of the restric-

tion σ(X,m)∗(A), as can be seen by the expressions of the pushforwards

as colimits over surjective maps; then the composition of pushforwards

from ∆X,m to ∆X,n and then to ∆X , is the same as the pushforward

σ(X,m)! to give the claimed formula.

Lemma 15.1.1 The functor skm preserves unitality, that is if A ∈

PC(X,M ) then skm(A) ∈ PC(X,M ). It has a right adjoint cskm :

PC(X,M ) → PC(X,M ) called the m-coskeleton. Both of these are

compatible with changing X so they give functors PC(M ) → PC(M ).

The natural morphism induces an isomorphism colimmskm(A) ∼= A. The

morphisms skm−1(A) → skm(A) → A are monomorphisms if M is a

presheaf category.

Proof Suppose (x0) is a sequence of length zero. Then (x0) ∈ ∆X,m for

any m ≥ 0 so the category of surjections (x0) ։ y· considered above has

as initial object (x0) itself. The formula (15.1.1) then says

skm(A)(x0) = A(x0).

It follows that if A is unital, so is skm(A).

The functors σ(X,m)! and σ(X,m)∗ have right adjoints σ(X,m)∗

and σ(X,m)∗ respectively. So, considered as a functor between diagram

categories, skm has as right adjoint:

cskm(B) = σ(X,m)∗σ(X,m)∗(B).

As before we can note that this functor has the expression

cskm(B)(x·) = limy· →x·

B(y·)

where the limit is taken over inclusions y· → x· in ∆X such that y· is

in ∆X,m i.e. has length ≤ m. Then again note that if x· = (x0) is a

sequence of length zero, the index category for the limit has an initial

object (x0) itself, so

cskm(B)(x0) = B(x0).

Thus cskm preserves the unitality condition and gives an endofunctor of

PC(X,M ), right adjoint to skm.

Suppose ψ : X → Y is a morphism of sets. For any A ∈ PC(X,M )


and B ∈ PC(Y,M ) together with a morphism f : A → ψ∗B inPC(X,M )

we would like to get a map skm(A) → ψ∗skm(B). In view of the defi-

nition of skm this is equivalent to giving a map of diagrams over ∆X,m

σ(X,m)∗(A) → σ(X,m)∗ψ∗skm(B). (15.1.2)

Let ψm : ∆X,m → ∆Y,m denote the induced functor, then ψσ(X,m) =

σ(Y,m) ψm so

σ(X,m)∗ψ∗skm(B) = ψ∗mσ(Y,m)∗skm(B)

= ψ∗mσ(Y,m)∗(B) = σ(X,m)∗ψ∗(B)

and functoriality for σ(X,m)∗ applied to f gives a map

σ(X,m)∗(f) : σ(X,m)∗(A) → σ(X,m)∗ψ∗(B).

Which yields the desired map (15.1.2). We also have a commutative

square

skm(A) → ψ∗skm(B)

A↓

→ ψ∗B.

↓

So, given a morphism A → B in PC(M ) this construction has given

skm(A) → skm(B). It is compatible with identities and compositions so

skm is an endofunctor ofPC(M ), with natural transformation skm(A) → A

again giving a structure of monadic projection to the full subcategory

of m-skeletic precategories.

Similarly for the coskeleton we look for a map

ψ!cskm(A) → σ(Y,m)∗σ(Y,m)∗(B). (15.1.3)

This is equivalent to looking for

σ(Y,m)∗ψ!cskm(A) → σ(Y,m)∗(B).

But in general

ψ!(C)(y·) = colimy· → ψx·C(x·)

and the colimit can be taken over surjective maps y· ։ ψx·. Thus if y·has length m then it only depends on the restriction of C to ∆X,m, in

particular

σ(Y,m)∗ψ!cskm(A) = σ(Y,m)∗ψ!(A)

304 Cofibrations

and we can look for a map ψ!A → cskm(B) or equivalently by adjunction,

A → ψ∗cskm(B). But the natural map B → cskm(B) gives by pullback

under ψ and composition

A → ψ∗(B) → ψ∗cskm(B)

as required. We get the functoriality maps for defining

cskm : PC(M ) → PC(M ),

again with a natural transformation A → cskm(A).

Suppose A ∈ PC(X,M ). For any x· a sequence of length p, then for

m ≥ p we have skm(A)(x·) = A(x·). Hence the morphism

colimmskm(A) → A

is an isomorphism on each object x· of ∆X . As colimits in diagram

categories are computed objectwise and the unitalification operation U!

preserves colimits, we have A ∼= colimmskm(A) in PC(X,M ). Recall

from Corollary 12.6.2 that connected colimits in PC(X,M ) also are

colimits in PC(M ) so we get the same formula in PC(M ).

One can remark that dually and for the same reason,

A∼=→ lim

mcskm(A).

Finally we note that if M is a presheaf category then we have a family

of functors hu : M → Set, the evaluations on objects of the underlying

category for the presheaves, such that hu preserves colimits. It follows

that hu(skm(A)) = skm(huA), however huA is a set-valued diagram

over ∆X . As is well-known, the inclusion of them-skeleton is an injection

of simplicial sets, and this works equally well for ∆oX -sets. Thus, the map

hu (skm(A)) = skm(hu A) → hu A

is a monomorphism. Therefore the map skm(A) → A induces a monomor-

phism upon application of all of the functors hu; but as M is a presheaf

category this implies that our map is a monomorphism as claimed in the

last statement of the lemma.

Notice that sk0(A) = disc(Ob(A)) is the set Ob(A) considered as a

discrete precategory, that is one whose p-fold morphism objects are all

∗ for constant sequences or ∅ otherwise (see Section 12.5). Indeed if x·is a sequence then there is a surjection to a unique (y0) of length 0, if

and only if x· is constant, so sk0(A)(x·) = A(y0) = ∗ if x· is constant of

value y0, and sk0(A)(x·) = ∅ if x· is nonconstant.


If A ∈ PC(M ) and (x0, . . . , xm) is a sequence of objects with m ≥ 1,

define the degenerate subobject

d(A;x0, . . . , xm) := skm−1(A)(x0, . . . , xm).

It has a map denoted

δ(A;x0, . . . , xm) : d(A;x0, . . . , xm) → A(x0, . . . , xm),

and we can express D as a colimit:

Lemma 15.1.2 With the above notations,

d(A;x0, . . . , xm) = colimx·→y·A(y0, . . . , yp)

where the colimit is taken over surjective maps x· → y· in ∆Ob(A) such

that y· are sequences of length p < m. The map δ(A;x0, . . . , xm) is

obtained from the restriction maps of A via the universal property of

the colimit. The map δ(A;x0, . . . , xm) is a monomorphism if M is a

presheaf category.

Proof This comes from the definition of D and the colimit expres-

sion for skm−1. The monomorphism property comes from the previous

lemma.

15.2 Some natural precategories

Consider the ordered sets [k] ∈ Ob(∆) with the notation

[k] = υ0, . . . , υk, υ0 < · · · < υk.

For anyB ∈M and any [k] ∈ Ob(∆) define the precategory h([k];B) ∈

PC([k],M ) as follows. Let tk ∈ Ob(∆[k]) denote the tautological object

tk := (υ0, . . . , υk) of length k, and let

itk : tk → ∆[k]

denote the inclusion of the discrete category on a single point tk ∈

Ob(∆[k]). Let Btk denote the constant diagram with value B on the

one-point category tk, and put

h([k];B) := U!itk!(Btk).

Recall that U! is the unitalization operation, necessary here because the

∆X -diagram itk!(Bt[k]) will not in general be unital. The following

lemma gives a concrete description of h([k];B) and could be taken as its

definition.

306 Cofibrations

Lemma 15.2.1 Suppose (y0, . . . , yp) is any sequence of elements of

the set [k] with yj = υij . Then:

—if (y0, . . . , yp) is increasing but not constant i.e. ij−1 ≤ ij but i0 < ipthen

h([k];B)(y0, . . . , yp) = B;


h([k];B)(y0, . . . , yp) = ∗;

and otherwise, that is if there exists 1 ≤ j ≤ p such that ij−1 > ij then

h([k];B)(y0, . . . , yp) = ∅.

The pullback maps giving h([k];B) a structure of diagram are all either

the unique maps of the form ∅ → B, ∅ → ∗, or B → ∗, or the identity

B → B.

Proof The functor B 7→ h([k];B) is by construction left adjoint to the

functor itk∗ from PC([k],M ) to M which sends A to A(υ0, . . . , υk).

On the other hand we can check by hand (as in the proof of the next

lemma below) that the functor sending B to the precategory defined

explicitly in the statement of the lemma, is also adjoint to the same

functor.

Lemma 15.2.2 If B ∈ M and C ∈ PC(M ) then a morphism f :

h([k], B) → C is the same thing as a sequence of objects x0, . . . , xk ∈

Ob(C) together with a map ϕ : B → C(x0, . . . , xk) in M .

Proof Use the explicit description of the previous lemma. Given f we

get xi := f(υi) and the map ϕ is given by fυ0,...,υk. On the other hand,

given x· and ϕ, the restrictions C(x0, . . . , xk) → C(xi0 , . . . , xip) lead to

maps

B → C(xi0 , . . . , xip)

for any sequence as in the first part of the previous lemma; if i0 = · · · = ipthen this map factors through ∗ by the unitality condition for C, treating

the second part of the previous lemma; and nothing is needed for defining

a map in the third case of the previous lemma. This defines the required

map f .

Note that the above construction is functorial in B, that is a map


A → B induces h([k];A) → h([k];B). Define the “boundary” of h([k];B)

by the skeleton operation:

h(∂[k];B) := skk−1h([k];B),

with the natural inclusion

h(∂[k];B) → h([k];B).

It is also functorial in B.

Lemma 15.2.3 This boundary object has the following concrete de-

scription:

—if (y0, . . . , yp) is increasing but not constant i.e. ij−1 ≤ ij but i0 < ip,

and if there is any 0 ≤ m ≤ k such that ij 6= m for all 0 ≤ j ≤ k, then

h(∂[k];B)(y0, . . . , yp) = B;


h(∂[k];B)(y0, . . . , yp) = ∗;

and otherwise, that is if either there exists 1 ≤ j ≤ p such that ij−1 > ijor else if the map j 7→ yj is a surjection from 0, . . . , p to [k], then

h(∂[k];B)(y0, . . . , yp) = ∅.

Proof Use the description of Lemma 15.2.1 and the formula

h(∂[k];B)(y·) = colimy·։z·h(B)(z·).

If f : A → B is a cofibration in M , put

h([k], ∂[k];Af→ B) := h([k];A) ∪h(∂[k];A) h(∂[k];B).

We therefore obtain two natural maps coming from f , the first is

P ([k]; f) : h([k];A) → h([k];B)

and the second is

R([k]; f) : h([k], ∂[k];Af→ B) → h([k];B).

The maps P ([k]; f) will form the generators for the projective cofibra-

tions, while the R([k]; f) form the generators for the Reedy cofibrations.

Unfortunately we don’t have an easy way to describe generators for the

injective cofibrations.

308 Cofibrations

15.3 Projective cofibrations

The different model structures are characterized and differentiated by

their notions of cofibrations. In the projective structure, the generating

set is easiest to describe but on the other hand there is no easy criterion

for being a cofibration.

A map f : A → B is a projective cofibration if on the set of objects

it is an injective map of sets Ob(f) : Ob(A) → Ob(B), and if the map

Ob(f)!(A) → B is a cofibration in the projective model category struc-

ture PCproj(Ob(B),M ). Recall that Ob(f)!(A) is the precategory A

transported to the subset f(Ob(A)) ⊂ Ob(B), then extended by adding

on the discrete or initial precategory over the complementary subset

Ob(B)−Ob(f)(Ob(A)).

Say that f is a projective trivial cofibration if it is a projective cofi-

bration, and a global weak equivalence.

For now, we say that a morphism u : U → V in PC(M ) is a projective

fibration if it satisfies the right lifting property with respect to all pro-

jective trivial cofibrations; we say that u is a projective trivial fibration

if it is a projective fibration and a global weak equivalence.

For the time being we also need a separate notation: say that u :

U → V in PC(M ) is an apparent projective trivial fibration if it satisfies

the right lifting property with respect to all projective cofibrations. One

of our tasks in future chapters will be to identify the class of projective

trivial fibrations with the apparent ones. For now we can characterize

and use the apparent ones.

Lemma 15.3.1 A morphism u : U → V in PC(M ) is an appar-

ent projective trivial fibration if and only if Ob(u) is surjective and

U → Ob(u)∗(V ) is an objectwise trivial fibration. A morphism f :

A → B is a projective cofibration if and only if it satisfies the left lifting

property with respect to the apparent projective trivial fibrations.

Proof Consider the class A of morphisms u such that Ob(u) is sur-

jective and U → Ob(u)∗(V ) is an objectwise trivial fibration. If f is a

projective cofibration then it breaks down as a composition f = f ′ d

where f ′ : Ob(f)!(A) → B is a morphism in PC(Ob(B),M ) and d is the

extension by adding on the discrete precategory Ob(B)−Ob(f)(Ob(A)).

Similarly, if u ∈ A then u = p u′ where u′ : U → Ob(u)∗(V ) is an

objectwise trivial fibration in PC(Ob(U),M ) and p is the tautological

map Ob(u)∗(V ) → V . Notice that p satisfies the right lifting property

with respect to any morphism which induces an isomorphism on sets of

15.3 Projective cofibrations 309

objects, and also with respect to extensions by adding discrete sets; and

dually f ′ satisfies the left lifting property with respect to tautological

maps such as p, while d satisfies the left lifting property with respect to

any map surjective on objects. All told, the lifting property with f on

the left and u on the right, is equivalent to the lifting property with f ′

on the left and u′ on the right. This holds whenever u′ is an objectwise

trivial fibration and f ′ is a projective cofibration in PC(Ob(B),M ).

These considerations show that A is contained in the class of apparent

projective trivial fibrations. Furthermore, if u satisfies right lifting for

any projective cofibration f then in particular Ob(u) must be surjective

(using for f any extension by a nonempty discrete set); and u′ must

satisfy right lifting with respect to projective trivial cofibrations inducing

isomorphisms on sets of objects, so u′ is an objectwise trivial fibration

(by the projective model structure on the PC(X,M )). This shows that

the apparent projective trivial cofibrations are contained in A so the

two classes coincide.

Then, similarly, if f satisfies left lifting with respect to A then first of

all it must be injective on sets of objects, as seen by considering u which

are surjective maps of codiscrete objects (i.e. objects U with U(y·) = ∗

for all sequences y·). And then decomposing f = f ′ d where d is the

extension by the complementary subset, we see that f ′ should be a pro-

jective cofibration in PC(Ob(B),M ). Thus f is a projective cofibration

by definition. But all projective cofibrations satisfy lifting with respect

to the apparent projective trivial fibrations, by the definition of this

latter class, and equality with the class A shows that the class of pro-

jective cofibrations is exactly that which satisfies left lifting with respect

to A .

We have stability under pushouts and retracts.

Corollary 15.3.2 Assume that M is a tractable left proper cartesian

model category. Suppose f : A → B and g : A → C are morphisms in

PC(M ) such that f is a projective cofibration. Then the map

u : C → B ∪A C

is a projective cofibration. Furthermore the projective cofibrations are

stable under retracts and transfinite composition. The class of projective

trivial cofibrations is closed under transfinite composition and retracts.

Proof Stability under pushouts, retracts and transfinite composition

come from the characterization of the projective cofibrations as dual to

310 Cofibrations

the class of apparent projective trivial fibrations. The last sentence now

follows immediately from Proposition 14.6.4.

The advantage of the notion of projective cofibration is that the gen-

erating set is very easy to describe.

Proposition 15.3.3 Fix a generating set I for the cofibrations of M .

Then for any k and any f : A → B in I, consider the map

P ([k]; f) : h([k];A) → h([k];B).

The collection of these for integers k and all f ∈ I, together with the

map ∅ → ∗, forms a generating set for the class of projective cofibrations

in PC(M ).

Proof Any projective cofibration f : A → B factors as f = f ′d where

Ad→ A′ ⊔ disc(Z)

f ′

→ B

where Z = Ob(B) = Ob(f)(Ob(A)), where A′ = Ob(f)!(A) is the

precategory on the set Ob(f)(Ob(A)) obtained by transport of structure,

disc(Z) is the discrete precategory on the set Z, and d is the extension

morphism. In this situation furthermore, f ′ is a projective cofibration in

PC(X,M ) where X = Ob(B) = Ob(A′ ⊔ disc(Z)).

Now d is obtained by successive pushout along ∅ → ∗, while the

w!P ([k]; f) for various maps w : [k] → X form the set of generators

for the projective cofibrations in PC(X,M ). Thus, f ′ is a retract of a

transfinite pushout along the w!P ([k]; f). Putting these together gives

the expression of f as a retract of a transfinite pushout of morphisms in

our generating set.

15.4 Injective cofibrations

It is easier to see whether a given map is an injective cofibration, since

this condition is defined objectwise, but the only way to get a generating

set in general is by an accessibility argument.

A morphism f : A → B is an injective cofibration if Ob(f) is an

injective map of sets, and if the map Ob(f)!(A) → B is a cofibration in

the injective model category structure PCinj(Ob(B),M ). Since injective

cofibrations are defined objectwise, we can be more explicit about this

condition: it means that for any sequence (x0, . . . , xp) in Ob(A), the

map A(x0, . . . , xp) → B(f(x0), . . . , f(xp)) is a cofibration in M , for

15.4 Injective cofibrations 311

any constant sequence (y, . . . , y) at a point y ∈ Y − f(X) the map

∗ → B(y, . . . , y) is a cofibration in M , and for any non-constant sequence

(y0, . . . , yp) such that at least one of the yi is not in f(X), the map

∅ → B(y0, . . . , yp) is a cofibration i.e. B(y0, . . . , yp) is a cofibrant object.

Say that f is an injective trivial cofibration if it is an injective cofibra-

tion, and a global weak equivalence.

We again have the stability proposition:

Proposition 15.4.1 Assume that M is a tractable left proper carte-

sian model category. Suppose f : A → B and g : A → C are morphisms

in PC(M ) such that f is an injective cofibration. Then the map

C → B ∪A C

is an injective cofibration. Furthermore the injective cofibrations are sta-

ble under retracts and transfinite composition. The class of injective triv-

ial cofibrations is closed under transfinite composition and retracts.

Proof The explicit conditions given in the paragraph defining the in-

jective cofibrations, are defined objectwise over ∆Z where Z is the set

of objects of the target precategory. The conditions are preserved by

pushouts, retracts and transfinite composition. Again, the last sentence

now follows immediately from Proposition 14.6.4.

Lemma 15.4.2 Suppose M is a tractable model category. Then the

class of injective cofibrations in PC(M ) admits a small set of genera-

tors.

Proof Follow the argument given in Barwick [16] for the proof using a

general accessibility argument.

Unfortunately, we don’t get very much information on the set of gen-

erators. In Section 15.7 below, if M satisfies some further hypotheses

which hold for presheaf categories, then the injective cofibrations are

the same as the Reedy cofibrations and the generators will be described

explicitly.

A similar problem with the injective structure is that if M is tractable,

we don’t know whether the generating cofibrations for PC(M ) have

cofibrant domains. This question for diagram categories isn’t treated by

Lurie in [153] and seems to remain an open question.

For our purposes this question is one further reason for introducing

the Reedy model structure which has an explicit set of generators; in

many cases (such as when M is a presheaf category, Proposition 15.7.2)

the Reedy and injective structures will coincide.

312 Cofibrations

15.5 A pushout expression for the skeleta

The main observation crucial for understanding the Reedy cofibrations,

is an expression for the successive skeleta as pushouts along maps of the

form R([k]; f). If x0, . . . , xm is a sequence of objects, recall the notation

d(A;x0, . . . , xm)δ(A,x·)→ A(x0, . . . , xm)

from Section 15.2. The identity map

d(A;x·) == skm−1(A)(x0, . . . , xm)

corresponds by the universal property of h([m]; ·) to a map

h([m];d(A;x·)) → skm−1(A)

in PC(M ). On the other hand, by the definition of h(∂[m]; ·) and func-

toriality of the skeleton operation the map

h([m];A(x·)) → A

yields a map

h(∂[m];A(x·)) → skm−1(A).

These two maps agree on h(∂[m];d(A;x·)) so they give a map defined

on the coproduct

h([m], ∂[m];d(A;x·)δ(A;x·)−→ A(x·))

= h([m];d(A;x·)) ∪h(∂[m];d(A;x·)) h(∂[m];A(x·)),

giving the top map in the commutative square

h([m], ∂[m];d(A;x·)→ A(x·)) → skm−1(A)

h([m];A(x·))

R([m]; δ(A;x·))

↓

→ skm(A).

↓

The map on the bottom is given by adjunction from the equality

A(x0, . . . , xm) = skm(A)(x0, . . . , xm).

Putting these together over all sequences x0, . . . , xm of length m we get

an expression for skm(A).


Proposition 15.5.1 For any m we have an expression of skm(A) as

a pushout of skm−1(A) by copies of the standard maps R([m]; ·) indexed

by sequences x· = (x0, . . . , xm):

skm(A) = skm−1(A) ∪∐

x·h([k],∂[k];δ(A;x·))

∐

x·

h([m],A(x·)).

This is a pushout, and uses coproducts, in the category PC(M ).

Proof This is a classical fact about simplicial objects.

15.6 Reedy cofibrations

Consider a map f : A → B in PC(M ), giving for each k a map on

skeleta skk(A) → skk(B). Define the relative skeleton of f

A ∪skk(A) skk(B)skrel

k (f)→ B.

We say that f is a Reedy cofibration if the relative skeleton maps are

injective cofibrations for every k.

Lemma 15.6.1 The class of Reedy cofibrations is closed under pushout,

transfinite composition, and retracts.

Proof The skeleton operation is given by a pushforward which is a kind

of colimit, so it commutes with colimits over connected categories which

are computed levelwise. The relative skeleton map of a retract is again

a retract so closure under retracts comes from the same property in M

levelwise.

Theorem 15.6.2 Suppose f : A → B is map such that Ob(f) is

injective and view Ob(A) as a subset of Ob(B). The following conditions

are equivalent:

(a)—f is a Reedy cofibration;

(b)—for any m ≥ 1 the map

skm(A) ∪skm−1(A) skm−1(B) → skm(B) (15.6.1)

is an injective cofibration;

(c)—for any sequence (x0, . . . , xp) of objects in Ob(A), the map

A(x0, . . . , xp) ∪d(A;x0,...,xp) d(B;x0, . . . , xp) → B(x0, . . . , xp) (15.6.2)

is a cofibration in M , and for any sequence (y0, . . . , yp) of objects in

Ob(B) not all in Ob(A), the map d(B; y0, . . . , yp) → B(y0, . . . , yp) is a

314 Cofibrations

cofibration in M ;

(d)—letting X := Ob(B), the map Ob(f)!(A) → B is Reedy cofibrant in

the model structure PCReedy(X,M ) of Theorem 14.3.2.

Proof First note that (a) implies (c), indeed (c) is the statement of (a)

for k = p− 1 at the sequence of objects (x0, . . . , xp) or (y0, . . . , yp).

Next we show that (b) implies the following more general statement:

for any 0 ≤ n ≤ m the map

skm(A) ∪skn(A) skn(B) → skm(B) (15.6.3)

is an injective cofibration. This is tautological for m = n. Let n be fixed,

and suppose we know this statement for some m ≥ n. Then

skm+1(A)∪skn(A) skn(B) = skm+1(A)∪

skm(A) (skm(A)∪skn(A) skn(B)).

Injective cofibrations are stable under pushout (Lemma 15.4.2), and our

inductive hypothesis says that

skm(A) ∪skn(A) skn(B) → skm(B)

is an injective cofibration. Take the pushout of this by skm+1(A) over

skm(A) and use the previous identification, to get that

skm+1(A) ∪skn(A) skn(B) → skm+1(A) ∪

skm(A) skm(B)

is an injective cofibration. On the other hand condition (b) says that

skm+1(A) ∪skm(A) skm(B) → skm+1(B)

is an injective cofibration, so composing these gives that

skm+1(A) ∪skn(A) skn(B) → skm+1(B)

is an injective cofibration. This proves by induction that the maps (15.6.3)

are injective cofibrations.

This statement now implies condition (a), indeed if (x0, . . . , xp) is any

sequence of objects in Ob(B) then for any m ≥ p, m ≥ k we have

B(x0, . . . , xp) = skm(B)(x0, . . . , xp).

The same is true for A if the xi are all in Ob(A). Hence

A ∪skk(A) skk(B)(x0, . . . , xp) = skm(A) ∪skk(A) skk(B)(x0, . . . , xp),

so the map

A ∪skk(A) skk(B)(x0, . . . , xp) → B(x0, . . . , xp)


is the same as the map

skm(A) ∪skk(A) skk(B)(x0, . . . , xp) → B(x0, . . . , xp).

This latter is just the value of (15.6.3) on the sequence (x0, . . . , xp) so it

is a cofibration in M , as needed to show the Reedy condition (a). This

shows that (b) implies (a).

To complete the proof we need to show that (c) implies (b). For this,

use the expression of Proposition 15.5.1 for skm(B) as a pushout of

skm−1(B) and the standard inclusions R([m], δ(B; y0, . . . , yp)) and simi-

larly for A. We have a diagram

skm−1(A) → skm(A)

skm−1(B)

↓

→ skm(B)

↓

where the top arrow is pushout along the R([m], δ(A;x0, . . . , xp)) for

sequences of objects (x0, . . . , xp) of A, and the bottom arrow is pushout

along the R([m], δ(B;x0, . . . , xp)) for sequences of objects (x0, . . . , xp) of

B. Taking the pushout of the upper left corner of the diagram gives the

expression

skm(A) ∪skm−1(A) skm−1(B) =

skm−1(B) ∪∐

x·h([m],∂[m];δ(A;x·))

∐

x·

h([m],A(x·)).

The coproducts are over sequences x· = (x0, . . . , xm) of length m of

objects ofA, however it can be extended to a coproduct over sequences of

objects of B by setting A(x0, . . . , xm) := ∅ as well as d(A;x0, . . . , xm) :=

∅ if any of the xi are not in Ob(A). On the other hand,

skm(B) = skm−1(B) ∪∐

x·∂h([m],∂[m];δ(B;x·))

∐

x·

h([m],B(x·)).

Putting these two together, we conclude that

skm(B) =(skm(A) ∪skm−1(A) skm−1(B)

)∪∐

x·C(x·)

∐

x·

h([m],B(x·))

where

C(x·) := h([m],A(x·)) ∪h([m],∂[m];δ(A;x·)) h([m], ∂[m]; δ(B;x·)).

Hence, to prove that the map skm(A) ∪skm−1(A) skm−1(B) → skm(B)

316 Cofibrations

is an injective cofibration, using stability of injective cofibrations under

pushouts, it suffices to show that each of the maps

C(x0, . . . , xm) → h([m],B(x0, . . . , xm)) (15.6.4)

is an injective cofibration. For both sides, the set of objects is now our

standard set [m] = υ0, . . . , υm. Consider the value on a sequence of

objects (y0, . . . , yp) in [m]. There are several possible cases:

(i)—If it is a constant sequence, then

h([m],A(x·))(y·) = h([m], ∂[m]; δ(A;x·))(y·)

= h([m], ∂[m]; δ(B;x·))(y·) = h([m],B(x·))(y·) = ∗

and the map (15.6.4) is the identity.

(ii)—If the sequence is somewhere decreasing i.e. there is some j with

yj < yj−1 then

h([m],A(x·))(y·) = h([m], ∂[m]; δ(A;x·))(y·)

= h([m], ∂[m]; δ(B;x·))(y·) = h([m],B(x·))(y·) = ∅

and again the map (15.6.4) is the identity.

(iii)—If the sequence is nondecreasing, but there is some υi not contained

in y· then

h([m],A(x·))(y·) = h([m], ∂[m]; δ(A;x·))(y·) = A(x·),

and

h([m],B(x·))(y·) = h([m], ∂[m]; δ(B;x·))(y·) = B(x·),

so in this case

C(x·)(y·) = A(x·) ∪A(x·) B(x·) = B(x·)

and once again (15.6.4) is the identity.

(iv)—If the sequence is nondecreasing and surjects onto the full set of

objects, then

h([m],A(x·))(y·) = A(x·), h([m], ∂[m]; δ(A;x·))(y·) = d(A;x·),

and

h([m],B(x·))(y·) = B(x·), h([m], ∂[m]; δ(B;x·))(y·) = d(B;x·),

so

C(x·)(y·) = A(x·) ∪d(A;x·) d(B;x·)


so the map

C(x·)(y·) → h([m],B(x·))(y·)

is exactly the map (15.6.2)

A(x·) ∪d(A;x·) d(B;x·) → B(x·)

which is known to be a cofibration in M because we are assuming condi-

tion (c) of the theorem. This completes the proof that the map (15.6.1)

is an injective cofibration, showing (c)⇒(b). This completes the proof of

the equivalence of (a), (b) and (c).

For the equivalence with (d), note that skm(Ob(f)!A) = Ob(f)!skm(A)

by commutation of pushforwards. Now

A ∪skm(A) B = Ob(f)!(A) ∪Ob(f)!skm(A) B

from the definition of colimits in PC(M ) (Section 12.6), so A → B is

Reedy cofibrant if and only if Ob(f)!A → B is. However, this latter

induces an isomorphism on sets of objects, and for such maps the cri-

terion (c) above is the same as the Reedy condition that the relative

latching maps be cofibrant in M . This shows that (d) is equivalent to

(a),(b),(c).

The map (15.6.4) occuring above is of the form R([m], g) as shown in

the following lemma.

Lemma 15.6.3 Suppose

Ea→ F

U

u

↓b→ V

v

↓

is a diagram in M . Consider the induced map g : U ∪E F → V . Then

the two maps

h([m], U) ∪h([m],∂[m];Eu→U) h([m], ∂[m];F

u→ V ) → h([m], V )

and

R([m], g) : h([m], ∂[m]; g) → h([m], V )

are the same.

318 Cofibrations

Proof Look at the values on any sequence y· = (y0, . . . , yp) of objects

in [m] = υ0, . . . , υm. There are several possibilities:

—if the sequence is constant then both maps are ∗ → ∗;

—if the sequence is anywhere decreasing then both maps are ∅ → ∅;

—if the sequence is nondecreasing but misses some element υj, then we

are in the boundary ∂[m] and the first map is

U ∪U V → V

and the second map is U → V , these are the same;

—if the sequence is nondecreasing and surjects onto [m] then both maps

are

U ∪E F → V.

We need to point out that these identifications of the maps, and in

particular of their sources, are functorial in the restriction maps for

diagrams over ∆o[m], so they give an identification

h([m], U) ∪h([m],∂[m];Eu→U) h([m], ∂[m];F

u→ V ) ∼= h([m], ∂[m]; g)

and both maps from here to h([m], V ) are the same.

Corollary 15.6.4 For an object A ∈ PC(M ), the following are equiv-

alent:

(a)—A is Reedy cofibrant;

(b)—for any m ≥ 1 the map skm−1(A) → skm(A) is an injective cofi-

bration;

(c)—for any sequence of objects (x0, . . . , xp) the map

skp−1(A)(x0, . . . , xp) → A(x0, . . . , xp)

is a cofibration in M ; (d)—A is a Reedy cofibrant object in PCReedy(X,M )

where X = Ob(A).

Proof Apply the previous proposition to the map ∅ → A. Note that

Ob(∅) = ∅ and skm(∅) = ∅. Condition (c) here is the part of condition

(c) of the proposition, concerning sequences of objects not all coming

from the source.

Corollary 15.6.5 If f : A → B is a cofibration in M , then R([k]; f)

(page 307) is a Reedy cofibration.

Proof Recall that

h([k], ∂[k], f) = h([k], A) ∪skk−1h([k],A) skk−1h([k], B)


and R([k]; f) is the map from here to h([k], B). If m ≤ k − 1 then

skmh([k], ∂[k], f) = skm([k], B) so the map occuring in condition (b) of

Theorem 15.6.2 is the identity in this case. For m ≥ k,

skmh([k], ∂[k], f) ∪skm−1h([k],∂[k],f) skm−1h([k], B) =

skmh([k], A) ∪skm−1h([k],A) skm−1h([k], B)

so the map occuring in condition (b) is

skmh([k], A) ∪skm−1h([k],A) skm−1h([k], B) → skm(h[k], B).

As for the equivalence with condition (c), it suffices to note that this in-

duces a cofibration over sequences x0, . . . , xm of length m in υ0, . . . , υk

(similar to the proof of Proposition 15.7.1 below).

Corollary 15.6.6 Many maps between the h([k], B) are Reedy cofi-

brations due to the previous corollary. For example if f : A → B is a

cofibration in M then the map

h([k − 1], A) → h([k], B)

induced by applying f at one of the faces of the k-simplex [k − 1] ⊂ [k],


Proof Either calculate directly the skeleta, or use the previous corollary

inductively.

The following statement is somewhat similar to giving a set of gen-

erators for the Reedy cofibrations. On the one hand it refers to the full

class of morphisms of the form R([m], g) for cofibrations g, while on the

other hand giving a stronger expression without refering to retracts.

Proposition 15.6.7 A morphism f : A → B is a Reedy cofibration

if and only if it is a transfinite composition of a disjoint union with a

discrete set, and then pushouts along morphisms of the form R([m], g)

for cofibrations g in M .

Proof A Reedy cofibration f : A → B can be expressed as the countable

composition of the morphisms A ∪skm−1(A) skm−1(B) → A ∪skm(A)

skm(B) which are themselves Reedy cofibrations. At the start, A∪sk0(A)

sk0(B) is the disjoint union of A with the discrete set Ob(B)− Ob(A).

Then, as we have seen in the proof of Theorem 15.6.2, at each stage

320 Cofibrations

the morphism in question is obtained by simultaneous pushout along

morphisms (15.6.4) of the form

h([m], U) ∪h([m],∂[m];Eu→U) h([m], ∂[m];F

v→ V ) → h([m], V ) (15.6.5)

where from the notations of (15.6.4) we put U := A(x0, . . . , xm), V :=

B(x0, . . . , xm), E := d(A;x0, . . . , xm) and F := d(B;x0, . . . , xm), with

u := δ(A;x0, . . . , xm) : E → U

and

v := δ(B;x0, . . . , xm) : F → V.

The maps u and v, as well as the maps U → V and E → F all fitting

into a commutative square. Condition (c) of the theorem says that the

map g : U∪EF → V is a cofibration. By Lemma 15.6.3, the map (15.6.5)

is the same as R([m]; g).

The the notion of Reedy cofibration is similar to that of projective

cofibration, in that we can give explicitly the set of generators.

Lemma 15.6.8 Suppose I is a set of maps in M . Let R(I) ⊂ Arr(PC(M ))

denote the set of all arrows of the form R([k]; g) for k ∈ N and g ∈ I. If

f ∈ cell(I) and m ∈ N then R([m]; f) ∈ cell(R(I)).

Proof Look at the behavior of R([k]; f) under pushouts and transfinite

composition. Suppose

Af→ B

C

g

↓v→ P

u

↓

is a pushout diagram in M , that is P = B∪AC. This induces a diagram

h([k], ∂[k]; g) → h([k], ∂[k];u)

h([k];C)

R([k]; g)

↓

→ h([k];P ).

R([k];u)

↓

We claim that this second diagram is then also a pushout in PC(M ).

In fact it is a diagram in PC([k],M ), and connected colimits of di-

agrams in PC([k],M ) are the same as the corresponding colimits in


PC(M ), also in turn they are the same as the corresponding colimits

in Func(∆o[k],M )(Section 12.6), so the pushout of the second diagram

can be computed objectwise. Then it is easy to see that it is a pushout,

using the explicit description of the values of h(([k], ∂[k]); ·) and h([k]; ·).

The conclusion from this discussion, is that any pushout along R([k];u)

will also be a pushout along R([k]; g).

Consider now a transfinite composition: suppose we have a series

. . . → Aifi,i+1→ Ai+1 → . . .

in M indexed by i ∈ β for some ordinal β. To treat limit ordinals

we need also to consider the transition maps fi,j : Ai → Aj for any

i < j. Assume that if j is a limit ordinal then Aj = colimi<jAi, and let

Aβ := colimi∈βAi. Consider the map

f : A0 → Aβ .

Wewould like to expressR([k]; f) as a transfinite composition of pushouts

along R([k]; ·). Consider the series

Gi := h([k], ∂[k];Ai → Aβ),

. . . → Gigi,i+1→ Gi+1 → . . .

with the more general transition maps gi,j : Gi → Gj . This is still

a transfinite series: if j is a limit ordinal then Gj = colimi<jGi, and

Gβ := colimi∈βGi is equal to h([k], ∂[k]; 1Aβ) = h([k];Aβ). These can be

seen by calculating the colimits objectwise over ∆[k]. The map

G0 → Gβ

is equal to R([k]; f). Furthermore, Gi+1 is the pushout of Gi along the

map R([k]; fi,i+1). Thus, R([k];A0 → Aβ) is a transfinite composition of

pushouts along the R([k]; fi,i+1). In turn, if fi,i+1 is a pushout along an

element of I then by the discussion of pushouts above, R([k]; fi,i+1) is a

pushout along an element of R(I). Thus, if our series gives an expression

for f as an element of cell(I), then R([k]; f) is seen to be in cell(R(I)).

322 Cofibrations

A similar statement is needed for retracts. Suppose

B

Af→

g

→

C

p

↓

s

↑

is a retract diagram in M , that is f = pg, g = sf , and ps = 1. This

gives a diagram

h([k], ∂[k]; g)R([k]; g)

→ h([k];B)

h([k], ∂[k]; f)

↓

↑

R([k]; f)→ h([k];C)

↓

↑

where the vertical arrows are induced by p and s.

Lemma 15.6.9 In the above situation, suppose we are given a pushout

diagram

h([k], ∂[k]; f) → U

h([k];C)

R([k]; f)

↓

→ V .↓

Then U → V is a retract of the pushout of U along R([k]; g), in the

category of objects under U .

Proof Using the left downward map in the previous diagram, we get a

map h([k], ∂[k]; g) → U so we can form the pushout V ′ of U along the

map R([k]; g). The vertical maps on the right of the previous diagram

induce maps V → V ′ and V ′ → V , compatible with the maps from U ,

and the composition V → V ′ → V is the identity as desired. To see all

of these things, note that in

h([k], ∂[k]; g)

U ←←

h([k], ∂[k]; f)

↓

↑


both triangles, obtained by using the upward or the downward arrows,

commute. The required statements are then obtained by functoriality of

pushout diagrams using the identity on U .

Corollary 15.6.10 If f ∈ cof(I) then R([k]; f) ∈ cof (R(I)).

Proof Write f as a retract of g ∈ cell(I), and apply the previous

Lemma 15.6.9 to R([k]; f) seen as a pushout of itself and the identity;

thus R([k]; f) is a retract of a pushout along R([k]; g). On the other

hand, Lemma 15.6.8 shows that R([k]; g) ∈ cell(R(I)). Thus, R([k]; f)

is a retract of a map in cell(R(I)), so it is in cof(R(I)).

Proposition 15.6.11 Fix a generating set I for the cofibrations of M .

Then for any k and any f : A → B in I, consider the map

R([k]; f) : h(([k], ∂[k]);Af→ B) → h([k];B).

The collection of these for integers k and all f ∈ I, forms a generating

set R(I) for the class of Reedy cofibrations in PC(M ). If the elements of

I have cofibrant domains, then the elements of R(I) have Reedy-cofibrant

domains.

Proof We have seen in Proposition 15.6.7 that any Reedy cofibration

can be written as a successive pushout by maps of the form R([k]; f)

for various k and various cofibrations f in M . As cof(R(I)) is closed

under pushout and transfinite composition, is suffices to show that for

any cofibration f , the map R([k]; f) is in cof (R(I)). This is exactly

the statement of the previous Corollary 15.6.10. If furthermore f has

cofibrant domain then R([k]; f) will have a Reedy cofibrant domain.

If M is cartesian, then the Reedy cofibrations also satisfy the cartesian

property. This is one of the main reasons for introducing the Reedy

objects. This is closely related to the corresponding result for diagrams

over a Reedy category, see for example Barwick [17] and Berger-Moerdijk

[32]. I would like to thank several people including Clemens Berger, Clark

Barwick, Ieke Moerdijk, andMark Johnson, for replying to a query about

this on the topology mailing list.

Proposition 15.6.12 Suppose M is a cartesian model category. Con-

sider morphisms Af→ B and U

g→ V in M . Assume that they are

cofibrations. Then for any k,m the morphism ξ from

U := h([k], ∂[k]; f)×h([m];V )∪h([k],∂[k];f)×h([m],∂[m];g)h([k];B)×h([m], ∂[m]; g)

to F := h([k];B)× h([m];V ) is a Reedy cofibration.

324 Cofibrations

Proof Use the criterion (c) of Theorem 15.6.2. Consider any sequence

of objects z· = ((x0, y0), . . . , (xp, yp)) where xi ∈ [k] and yj ∈ [m]. We

first need to calculate skp−1(U)(z·) and skp−1(F)(z·).

If either one of the sequences x· or y· is decreasing at any index, then

(using that the product of anything with ∅ is again ∅)

skp−1(U)(z·) = skp−1(F)(z·) = U(z·) = F(z·) = ∅,

and the map (15.6.2) for ξ is an isomorphism hence a cofibration. Simi-

larly, if z· is constant then

skp−1(U)(z·) = skp−1(F)(z·) = U(z·) = F(z·) = ∗,

so again the map (15.6.2) is an isomorphism hence a cofibration.

Thus, we may assume that both sequences x· and y· are nondecreasing

and at least one of them is nonconstant.

However, if one or the other of the sequences is constant, then the

morphism in question becomes the same as the map (15.6.2) for the other

side, and we know that R([k]; f) or R([m]; g) are Reedy cofibrations by

15.6.4.

So, we may now assume that both sequences are nonconstant.

If z· is strictly increasing, then it has no quotient z· → w· of length

≤ p− 1, so

skp−1(F)(z·) = skp−1(U)(z·) = ∅.

So in this case the map (15.6.2) is just the map

U(z·) → F(z·).

First note that F(z·) = B × V . The calculation of U(z·) breaks into

several cases. If x· lies in ∂[k] (i.e. it misses at least one object of [k])

and y· lies in ∂[m] then

U(z·) = B × V ∪B×V B × V = B × V,

so the map from here to F(z·) is an isomorphism hence a cofibration. If

x· lies in ∂[k] but y· surjects onto [m] then

U(z·) = B × V ∪B×U B × U = B × V,

so again the map to F(z·) is an isomorphism. Similarly, if x· surjects

onto [k] but y· lies in ∂[m], then

U(z·) = A× V ∪A×V B × V = B × V,


and the map to F(z·) is an isomorphism. Finally, suppose that x· surjects

onto [k] and y· surjects onto [m]. Then

U(z·) = A× V ∪A×U B × U,

so the map from here to B × V is a cofibration by the cartesian axiom

for M and the assumption that f and g were cofibrations of M . This

completes the proof that the map (15.6.2) for ξ is a cofibration in the

case where the sequence z· is strictly increasing.

Assume therefore that z· is not strictly increasing, i.e. it has at least

one adjacent pair of objects which are equal. We have

skp−1(F)(z·) = d(F ; z·) = colimz· → w·F(w·)

skp−1(U)(z·) = d(U ; z·) = colimz· → w·U(w·)

where the colimits are taken over surjective maps z· → w· such that w·

has length less than or equal to p− 1, see Lemma 15.1.2. The category

of quotients z· → w· of length q ≤ p − 1 is nonempty, because we are

assuming that z· is not strictly increasing. The opposite category of this

category of quotients has an initial object, corresponding to the quotient

of minimal length obtained by identifying all adjacent equal objects.

The diagram which to z· → w· associates F(w·) is constant, taking

values B×V . This uses the definitions of h([k];B) and h([m];V ) and the

fact that both sequences x· and y· are nondecreasing and nonconstant.

Hence,

skp−1(F)(z·) = colimz· → w·B × V = B × V = F(z·)

since the colimit of a constant diagram over a category with initial ob-

ject, is equal to the constant value of the diagram.

Next, look at

skp−1(U)(z·) = d(U ; z·) = colimz· → w·U(w·).

Suppose given a quotient z· → w· of length q ≤ p − 1. Write w· =

((r0, s0), . . . , (rq, sq)). Note that r· is a quotient sequence of x· and s·is a quotient sequence of y·. The question of whether the sequence of

first elements r· lies in ∂[k] or [k], or whether the sequence of second

elements s· lies in ∂[m] or [m], is independent of the choice of quotients

and depends only on x· or y·. Hence, the values h([k], ∂[k]; f)(r·) and

h([m], ∂[m]; g)(s·) are independent of the choice of w·, and the colimit

defining skp−1(U)(z·) is equal to its constant value on any of the objects

326 Cofibrations

w·. This breaks into exactly the same cases as considered previously, and

by the same reasoning we see that

skp−1(U)(z·) = U(z·).

Now the map (15.6.2) for ξ is written as

U(z·) ∪skp−1(U)(z·) skp−1(F)(z·) → F(z·),

but in view of the identifications given above this map is just the identity

of F(z·) = B × V so it is a cofibration. This completes the proof of the

proposition.

Corollary 15.6.13 Suppose f : A → B and g : U → V are Reedy

cofibrations in PC(M ). Then

A× V ∪A×U B × U → B × V

is a Reedy cofibration in PC(M ).

Proof Both f and g may be expressed as transfinite compositions of

pushouts along elementary Reedy cofibrations of the form R([k], h). The

previous proposition gives the cartesian property for these. Since Reedy

cofibrations are closed under pushout and transfinite composition, we

get the cartesian property for any f and g.

One of the main steps in our proof will be to give the same property

for trivial Reedy cofibrations, in Chapter 19 below.

15.7 Relationship between the classes of cofibrations

Proposition 15.7.1 A projective cofibration is a Reedy cofibration,

and a Reedy cofibration is an injective cofibration.

Proof Starting from generators for the projective cofibrations, if f :

A → B is a cofibration in M then h([k], f) is a Reedy cofibration.

Indeed, the set of objects is υ0, . . . , υk. Suppose given a sequence of the

form x· = (υi0 , . . . , υim). If the sequence is increasing and nonconstant,

then the same is true of any surjective image, and we get

h([k], A)(x·) = A, skmh([k], A)(x·) = A or ∅

with ∅ occuring if there are no surjections to sequences of length ≤ m.

Similarly for B. The relative skeleton map for h([k], f) at x· is either f

or the identity of B in this case. If the sequence is constant, then the

15.7 Relationship between the classes of cofibrations 327

same is true of any surjective image and the relative skeleton map is the

identity of ∗. If the sequence is anywhere strictly decreasing, again the

same is true of any surjective image and the relative skeleton map is the

identity of ∅. Thus, h([k], f) is a Reedy cofibration.

Using m = 0 in the definition of Reedy cofibrations, we get that they

are levelwise cofibrations.

Proposition 15.7.2 Suppose M is a presheaf category and the cofi-

brations are the monomorphisms of M . Then the Reedy and injective

cofibrations of PC(M ) coincide, and if I is a generating set of cofibra-

tions for M then the set R(I) consisting of the R([k]; f) for f ∈ I, is a

generating set of cofibrations for the injective cofibrations.

Proof If M = Presh(Φ) we can verify the cofibrant property of the

relative skeleton maps levelwise over Φ. It then reduces to the classical

statement that the skeleton of a simplicial set is a simplicial subset.

Theorem 15.7.3 Suppose a map f : A → B in PC(M ) satisfies the

right lifting property with respect to the class of projective (resp. injective,

Reedy) cofibrations. Then f is a global weak equivalence.

Proof It suffices to treat the case of projective cofibrations, since the

other ones contain this class.

Recall that any set X ∈ Set corresponds to a discrete precategory

disc(X) ∈ PC(M ) whose object set is X itself, and whose morphism

objects are defined by

disc(X)(x0, . . . , xn) =

∗ if x0 = · · · = xn∅ otherwise.

Included among the projective cofibrations is ∅ → disc(x). The

morphism f : A → B in PC(M ) satisfies the right lifting property with

respect to ∅ → x, if and only if Ob(A) → Ob(B) is surjective. In

particular f is essentially surjective.

Next, suppose g is a generating cofibration of M . If f satisfies the right

lifting property with respect to a given h([k], g), Lemma 15.2.2 implies

that for any [k]-sequence of objects x0, . . . , xk ∈ Ob(A), the map

A(x0, . . . , xk) → B(f(x0), . . . , f(xk))

satisfies the right lifting property with respect to g. If f satisfies the right

lifting property with respect to the generators of projective cofibrations

of PC(M ) which is to say all the h([k], g) as g runs over a generating

328 Cofibrations

set of cofibrations of M and k is any positive integer, it follows that for

any x0, . . . , xk ∈ Ob(A), the map

A(x0, . . . , xk) → B(f(x0), . . . , f(xk))

is a trivial fibration in M—in particular it is a weak equivalence. This

shows that f is fully faithful.

16

Calculus of generators and relations

In this chapter we look more closely at the specific calculus of generators

and relations corresponding to the direct left Bousfield localization of

PC(X ;M ) discussed in Chapter 14. Throughout, the model category

M is assumed to be tractable left proper and cartesian.

16.1 The Υ precategories

Recall that ∅ denotes the initial object and ∗ the coinitial object of

M . We introduce some M -enriched precategories Υk(B1, . . . , Bk) via

an adjunction.

If X is a set, it may be considered as a discrete precategory disc(X)

with object set itself, and morphism objects disc(X)(x0, . . . , xn) := ∅

when some xi 6= xj but disc(X)(x0, . . . , xn) := ∗ when x0 = · · · = xn.

We can consider the category disc(X)/PC(M ) of arrows disc(X) → A

in PC(M ). Such an arrow is equivalent to giving an M -enriched pre-

category A together with a map of sets X → Ob(A).

Recall that [k] denotes the ordered set υ0, . . . , υk, so we can form

the category disc([k])/PC(M ).

Looking at the morphism objects between adjacent elements of [k]

gives a functor

(E1, . . . , Ek) : disc([k])/PC(M ) →M × · · · ×M

defined by

(f : [k] → Ob(A)) 7→ (A(f(0), f(1)),A(f(1), f(2)), . . . ,A(f(k − 1), f(k))) .

It has a left adjoint, consisting of a functor

Υk : Mk → PC(M )


330 Calculus of generators and relations

together with a natural transformation υ : disc([k]) → Υk(B1, . . . , Bk)

so that the resulting functor

Mk → disc([k])/PC(M ), (B1, . . . , Bk) 7→

(disc([k])

υ→ Υk(B1, . . . , Bk)

)

is left adjoint to (E1, . . . , Ek).

After this abstract introduction, we can describe explicitly Υ(B1, . . . , Bk),

and the reader could well skip the above discussion at first reading and

consider just the following construction. The main part of the structure

of precategory is given by

Υ(B1, . . . , Bk)(υi−1, υi) = Bi.

This is extended whenever there is a constant string of points on either

side:

Υ(B1, . . . , Bk)(υi−1, . . . , υi−1, υi, . . . , υi) = Bi.

The unitality condition on the diagram ∆oυ0,...,υk

→M implies, by

minimality of Υk, that for 0 ≤ i ≤ k we have

Υ(B1, . . . , Bk)(υi, . . . , υi) = ∗.

In all other cases, and

Υ(B1, . . . , Bk)(x0, . . . , xn) = ∅.

The reader is invited to check that the obvious maps turn Υ(B1, . . . , Bk)

as defined above, into a functor from ∆oυ0,...,υk

to M , which is unital,

in other words it is an element of PC(M ). The tautological map

υ : [k] → Ob(Υ(B1, . . . , Bk)) = υ0, . . . , υk, i 7→ υi

provides Υ(B1, . . . , Bk) with a structure of element of disc([k])/PC(M ).

The construction is clearly functorial in (B1, . . . , Bk), and υ is a nat-

ural transformation. We get a functor Υ : M k → disc([k])/PC(M ).

Lemma 16.1.1 The explicitly constructed Υ is left adjoint to the

functor (E1, . . . , Ek). Furthermore it satisfies a more global adjunction

property: if R ∈ PC(M ) then a morphism Υ(B1, . . . , Bk) → R is

the same thing as a string of objects x0, . . . , xk ∈ Ob(R), and maps

Bi → R(xi−1, xi).

Proof Given a string of objects x0, . . . , xk ∈ Ob(R), and mapsBi → R(xi−1, xi),

the functoriality maps for the diagram R : ∆oOb(R) →M provide the

required maps to define Υ(B1, . . . , Bk) → R and this is inverse to the

obvious construction in the other direction.


Corollary 16.1.2 There is also an expression as a pushout of the

standard representables:

Υ(B1, . . . , Bk) = h([1], B1) ∪υ1 h([1], B2) ∪

υ2 · · · ∪υk−1 h([1], Bk)

where the maps in the coproducts go alternately from the single point

precategories to the second or first objects of h([1],−).

Proof The pushout expression satisfies the same universal property as

given for Υ in the previous lemma.

The construction Υ sends sequences of cofibrations in M to Reedy

cofibrations.

Lemma 16.1.3 If fi : Ai → Bi are cofibrations in M then the induced

map

Υ(A1, . . . , Ak) → Υ(B1, . . . , Bk)


Proof Put together in a string the cofibrations R([1], fi) of Corollary

15.6.5.

One could form a more complicated Reedy cofibration out of the fi,

for example when k = 2

Υ(B1, A2) ∪Υ(A1,A2) Υ(A1, B2) → Υ(B1, B2)

is a Reedy cofibration. We leave it to the reader to elucidate notation

for the most general possibility.

As a particular case of the Υ construction which will enter into our

discussion in the next section, note that for (B1, . . . , Bk) = (B, . . . , B)

there is a tautological map Υ(B, . . . , B) → h([k], B). In terms of uni-

versal properties, if R ∈ PC(M ) and we are given a sequence of ob-

jects x0, . . . , xn ∈ Ob(R) and B → R(x0, . . . , xn), which corresponds

to h([k], B) → R, then composing with the principal edge maps (i.e.

those which make up the Segal map) we get B → R(xi−1, xi) and this

collection corresponds to the map Υ(B, . . . , B) → R.

16.2 Some trivial cofibrations

There is an important link between the pseudo-generating set used to

define global weak equivalences in Chapters 13 and 14, and the objects

Υ defined above.


A morphism of sets g : X → Y induces a functor ∆og : ∆o

X → ∆oY .

Consider a sequence of objects x0, . . . , xn ∈ X and the resulting Segal

functor P(x0,...,xn) : ǫ(n) → ∆oX . Then the composition with ∆o

g is equal

to the functor corresponding to the sequence g(x0), . . . , g(xn) ∈ Y :

∆og P(x0,...,xn) = P(g(x0),...,g(xn)) : ǫ(n) → ∆o

Y .

We obtain a diagram

Func(ǫ(n),M ) → Func(∆oX ,M )

g!→ Func(∆o

Y ,M )

PC(X,M )

↓g!→

→

PC(Y,M )

↓

where the vertical arrows are the unitalization operators U!.

If f : A → B is a generating cofibration for M , look at what happens

to the map

n(f) = (A,B, . . . , B; f, . . . , f)ρn(f)→ ξ0,!(B) = (B, . . . , B; 1, . . . , 1)

in Func(ǫ(n),M ) which was considered in Chapter 13. The image in

PC(X,M ) is the pseudo-generator

U!P(x0,...,xn),!(ρn(f)) : U!P(x0,...,xn),!(n(f)) → U!P(x0,...,xn),!(ξ0,!(B))

for the Reedy model structure considered in Theorem 14.3.2. When

we project to PC(Y,M ) using g! this gives the corresponding pseudo-

generator in PC(Y,M ) for the sequence g(x0), . . . , g(xn):

g!(U!P(x0,...,xn),!(ρn(f))

)= U!P(g(x0),...,g(xn)),!(ρn(f)).

This remark is particularly useful when X = [n] is the universal

set having a sequence of objects υ0, . . . , υn ∈ [n]. For any sequence

y0, . . . , yn ∈ Y there is a unique map g(y0,...,yn) : [n] → Y sending υito yi, and the standard generator for the cofibration f at the sequence

(y0, . . . , yn) is then expressed as

U!P(y0,...,yn),!(ρn(f)) = g(y0,...,yn),!(U!P(υ0,...,υn),!(ρn(f))

). (16.2.1)

So, in order to understand these generators it suffices to look at the

corresponding ones in PC([n],M ).

Using ζn(f) instead of ρn(f) gives the pseudo-generators for the pro-

jective model structure (Theorem 14.1.1). The pushouts by U!P(υ0,...,υn),!(ζn(f))


will not however have such a canonical description in view of the addi-

tional choice necessary to define ζn(f). The reader is invited to modify

the following discussion accordingly in that case.

Lemma 16.2.1 Fix n and suppose f : A → B is a cofibration in M .

Define the arrow

Ψ([n], f) :=(h([n], A) ∪Υ(A,...,A) Υ(B, . . . , B) → h([n], B)

)

in PC([n],M ). Then

U!P(υ0,...,υn),!(ρ(f)) = Ψ([n], f).

Proof The source of ρ(f) is the object (f) = (A,B, . . . , B; f, . . . , f)

in Func(ǫ(n),M ). Its image in Func(∆o[n],M ) is the universal object

for diagrams R with maps A → R(υ0, . . . , υn) and B → R(υi−1, υi) for

i = 1, . . . , n, making commutative diagrams

A → B

R(υ0, . . . , υn)

↓

→ R(υi−1, υi).

↓

After applying U! the image in PC([n],M ) is again the universal object

for unital diagrams R as above.

The collection of maps B → R(υi−1, υi) corresponds by adjunction to

a map Υ(B, . . . , B) → R, the map A → R(υ0, . . . , υn) corresponds to a

map h([n], A) → R, and the commutative diagrams amount to requiring

that they lead to the same map Υ(A, . . . , A) → R. Hence, our universal

diagram is the pushout

U!P(υ0,...,υn),!(ϕ(f)) = h([n], A) ∪Υ(A,...,A) Υ(B, . . . , B).

Similarly, the target of ρ(f) is the universal object for unital diagrams

S with maps B → S(υ0, . . . , υn), thus

U!P(υ0,...,υn),!(ξ0,!(B)) = h([n], B).

The map between them is the natural one induced by h([n], A) → h([n], B)

and Υ(B, . . . , B) → h([n], B).

Corollary 16.2.2 If y0, . . . , yn is any sequence of objects in Y and f

is a cofibration in M then U!P(y0,...,yn),!(ρ(f)) is obtained by applying

g(y0,...,yn),! to the map of the lemma.


Proof Apply (16.2.1) to the previous lemma.

Lemma 16.2.3 Suppose f : A → B is a cofibration in M and n ≥ 0.

Then Ψ([n], f) is a Reedy trivial cofibration in PC([n],M ) and a globally

trivial Reedy cofibration in PC(M ).

Proof This follows from Theorem 14.3.2.

For brevity, the source of Ψ([n], f) will be denoted

srcΨ([n], f) := h([n], A) ∪Υ(A,...,A) Υ(B, . . . , B).

The target is targΨ([n], f) = h([n], B).

Proposition 16.2.4 Suppose f : A → B is a cofibration in M , and

suppose R ∈ PC(M ). To give a map from the source of Ψ([n], f) to R

is the same as to give a sequence of objects x0, . . . , xn ∈ Ob(R) together

with a commutative diagram

A → B

R(x0, . . . , xn)

↓

→ R(x0, x1)× · · · × R(xn−1, xn)

↓

with f on the top and the Segal map on the bottom. The pushout R∪srcΨ([n],f)

h([n], B) computed in PC(M ) has a set of objects isomorphic to Ob(R)

and transporting by this identification (which will usually be made tac-

itly) this pushout is the same as the pushout of R along U!P(y0,...,yn),!(ρ(f))

in the category PC(Ob(R),M ).

Proof The description of maps srcΨ([n], f) → R was done in the proof

of Lemma 16.2.1. Since Ψ([n], f) is an isomorphism on sets of objects,

pushout along it preserves the set of objects up to canonical isomor-

phism.

In general if g : X → Y is a map of sets, if T → T ′ is a mor-

phism in PC(X,M ) and R ∈ PC(Y,M ) with a map g!(T ) → R, then

the pushout R∪T T ′ in PC(M ) corresponds (under the canonical iso-

morphism between Y and the pushout of Y along 1X) to the pushout

R ∪g!(T ) g!(T ′) in PC(Y,M ). Indeed they both satisfy the same uni-

versal property in PC(Y,M ) because for any R′ ∈ PC(Y,M ) a map

g!(T ) → R′ is the same thing as a map T → R′ in PC(M ) inducing g

on sets of objects; and the same for T ′.


Apply this general fact to pushouts along the map Ψ([n], f) to get the

last statement of the proposition.

Corollary 16.2.5 If A ∈ PC(M ) then there is a global trivial Reedy

cofibration A → Seg(A) obtained as a transfinite composition of pushouts

along morphisms either of the form Ψ([n], f), or levelwise trivial Reedy

cofibrations, such that Seg(A) satisfies the Segal condition.

Proof If X = Ob(A) then there is a transfinite composition of pushouts

along elements of the standard generating set for the direct left Bous-

field localization considered in Chapters 13 14, see Theorem 14.3.2. The

elements of the standard generating set are either generating trivial

cofibrations for the unital diagram theory without the product con-

dition, i.e. generating levelwise trivial projective cofibrations; or else

maps of the form U!P(x0,...,xn),!(ρ(f)). We have seen above that pushout

in PC(X,M ) along U!P(x0,...,xn),!(ρ(f)) is the same as pushout along

Ψ([n], f) in the global category PC(M ).

Notice from our discussion that any transfinite composition of pushouts

along maps as used in the corollary, is a global trivial cofibration, indeed

it is a trivial cofibration in the direct Bousfield localized projective model

structure constructed in Chapters 13 and 14. In particular, if A → A′′ is

some such transfinite composition, then applying the corollary to A′′ we

obtain another such transfinite composition A′′ → Seg(A′′) such that

Seg(A′′) satisfies the Segal condition. In this case all the maps

A → A′′ → Seg(A′′)

are global trivial cofibrations.

16.3 Pushout by isotrivial cofibrations

One of the main problems is to prove that pushout by a global weak

equivalence is again a global weak equivalence. An important first case

is pushout by a global weak equivalence which is an isomorphism on

objects. In the following discussion we use the generic term “cofibration”

for either a projective, Reedy or injective cofibration: the statements

come in three versions one for each of the model structures of Theorem

14.1.1 or Theorem 14.3.2.

An isotrivial cofibration is a cofibration Af→ B (in whichever of the

projective, Reedy, or injective structures we are using), such that Ob(f)


induces an isomorphism Ob(A) ∼= Ob(B), and f is a global weak equiv-

alence.

Lemma 16.3.1 A morphism Af→ B is an isotrivial cofibration, if and

only if Ob(f) is an isomorphism and Ob(f)!Af♯→ B is a trivial cofibra-

tion in the appropriate model structure of Theorem 14.1.1 or Theorem

14.3.2 on PC(X,M ), where X = Ob(B).

Proof The condition that Ob(f) is an isomorphism is tautologically

necessary so we assume it. By transport of structure using Ob(f)! we may

assume that f is a morphism in PC(X,M ). Considered as a morphism

of M -precategories f is then automatically essentially surjective. We get

the diagram

A → B

Seg(A)

↓

→ Seg(B)

↓

in PC(X,M ), where the vertical arrows are weak equivalences in the

model structures of 14.1.1 or 14.3.2. The fully faithful condition for f

is by definition the condition that the bottom map be a levelwise weak

equivalence; but this is equivalent to being a weak equivalence since the

objects satisfy the Segal conditions. By 3 for 2 this condition is equivalent

to the top map being a weak equivalence in PC(X,M ) for the model

structures of 14.1.1 or 14.3.2.

Lemma 16.3.2 Suppose A ∈ PC(M ) and f : B → C is an isotrivial

cofibration. Suppose furthermore that f is a levelwise weak equivalence

of diagrams, which means that for any sequence of objects (x0, . . . , xp) ∈

Ob(B) the map

B(x0, . . . , xp) → C(f(x0), . . . , f(xp))

is a weak equivalence in M . Suppose given a map g : B → A. Then

A → A∪B C

also induces an isomorphism on sets of objects, and is a levelwise weak

equivalence of diagrams. In particular, it is an isotrivial cofibration.

Proof By transport of structure we may assume that Ob(f) is the

identity of X = Ob(B) = Ob(C) and think of B, C ∈ PC(X,M ). Let


Y = Ob(A) and denote also by g : X → Y the map induced by g on

sets of objects. Then

A∪B C = A ∪g!(B) g!(C) in PC(Y,M )

but g!(B) → g!(C) is a levelwise trivial cofibration, so the pushout is a

levelwise trivial cofibration, hence in particular a trivial cofibration in

the model structure of Theorem 14.1.1 or Theorem 14.3.2.

Theorem 16.3.3 Suppose A ∈ PC(M ) and f : B → C is an isotriv-

ial cofibration (in the projective, Reedy or injective structures). Suppose

given a map g : B → A. Then

A → A∪B C

is an isotrivial cofibration (in the projective, Reedy or injective structures

respectively).

Proof Using Corollary 16.2.5 let

A → A′, B → B′, C → C′

be global trivial cofibrations towards objects which satisfy the Segal

property, obtained by series of pushouts along maps which are either

levelwise trivial cofibrations, or of the form Ψ([n], u) where u are gener-

ating cofibrations of M . Make the choice for B first, and letA′′ = A∪BB′

and C′′ = C ∪B B′. Then as in the remark after Corollary 16.2.5, we can

continue with A′′ → A′ and C′′ → C′. That way, there are maps

A′ ← B′ → C′

the second one still being a global trivial cofibration (in any of the pro-

jective, Reedy or injective structures), and

A ∪B C → A′ ∪B′

C′

is itself obtained by pushout along maps of the same form. In particular

this latter map is also a global trivial cofibration, so by the 3 for 2

property for global weak equivalences, it suffices to show that the map

A′ → A′ ∪B′

C′

is a global weak equivalence. But now, the map B′ → C′ satisfies the

hypothesis of Lemma 16.3.2, exactly by the fully faithful condition. So,

by that lemma, the map A′ → A′ ∪B′

C′ is a global trivial cofibration

as required.


Cofibrant pushouts are invariant under global weak equivalences in-

ducing isomorphisms on sets of objects.

Lemma 16.3.4 Suppose given a diagram

A ← B → C

A′

↓

← B′

↓

→ C′↓

such that the left horizontal arrows are injective cofibrations, and the

vertical arrows are global weak equivalences inducing isomorphisms on

sets of objects. Then the induced map

A ∪B C → A′ ∪B′

C′

is a global weak equivalence inducing an isomorphism on sets of objects.

Proof Choose a diagram

A ← B → C

A′′

↓

← B′′

↓

→ C′′↓

such that the vertical maps are injective isotrivial cofibrations, and the

precategories along the bottom row satisfy the Segal condition. We can

do this by first choosing B′′ := Seg(B) which is a transfinite composition

of pushouts along standard morphisms in the direct localizing system

Kinj of Theorem 14.1.1, which in the global context of PC(M ) may be

interpreted as the standard maps of Corollary 16.2.5. Let

A′′ := Seg(A ∪B B′′)

and similarly for C′′. That way the map B′′ → A′′ is again an injective

cofibration. Now put

B3 := Seg(B′ ∪B B′′),

then

A3 := Seg

(B3 ∪B

′∪BB′′

A′ ∪A A′′)

and

C3 := Seg

(B3 ∪B

′∪BB′′

C′ ∪C C′′).


We have the diagram

B′ ∪B B′′ → A′ ∪A (A ∪B B′′) → A′ ∪A A′′

B3

↓

→ B3 ∪B′∪BB′′

A′ ∪B B′′

↓

→ B3 ∪B′∪BB′′

A′ ∪A A′′

↓

B3

wwwwwwwwww→ Seg(. . .)

↓

→ A3

↓

in which the two upper squares are cartesian, the vertical maps are

obtained by pushouts along standard maps of Corollary 16.2.5, and the

top vertical arrows are injective cofibrations. It follows that B3 → A3

is again an injective cofibration. There is a similar diagram for C3 (but

the horizontal arrows are not necessarily cofibrant).

Furthermore, A3, B3 and C3 are pushouts of A′, B′ and C′ respec-

tively along the standard morphisms of Corollary 16.2.5. They fit into a

diagram

A′′ ← B′′ → C′′

A3

↓

← B3

↓

→ C3↓

consisting of objects satisfying the Segal condition. The vertical arrows

are global weak equivalences inducing isomorphisms on objects. Indeed

on the left for example, pushout along the isotrivial cofibration A → A′′

is again an isotrivial cofibration (Lemma 16.3.3), from which it follows

that the map A′ → A3 is a global weak equivalence. By 3 for 2, the map

A′′ → A3 is a global weak equivalence.

We get that the vertical arrows in the previous diagram are levelwise

weak equivalences of diagrams. These are preserved by pushout when

one of the maps in the pushout is a levelwise cofibration, as is verified

levelwise (recall that M is assumed to be left proper). Therefore the

map

A′′ ∪B′′

C′′ → A3 ∪B3

C3


is a levelwise weak equivalence of diagrams, so it is a global weak equiv-

alence.

Now, the map

A∪B C → A′′ ∪B′′

C′′

is obtained by a transfinite composition of pushouts along the standard

maps of Corollary 16.2.5, so it is a global weak equivalence. On the other

hand, the map

A′ ∪B′

C′ → A3 ∪B3

C3

is also obtained by a transfinite composition of pushouts along the stan-

dard maps, so it is a global weak equivalence. By 3 for 2 we conclude

that the map

A ∪B C → A′ ∪B′

C′

is a global weak equivalence.

A similar argument gives closure under transfinite composition. A

cofibrancy condition can be avoided in PC(M ) under the condition

that it could be avoided in M already.

Lemma 16.3.5 The notion of global weak equivalence in PC(M ) is

closed under transfinite compositions such that the transition maps are

any kind of cofibrations (injective ones being the weakest). If weak equiv-

alences of M are closed under transfinite composition, then the notion

of global weak equivalence in PC(M ) is also closed under transfinite

composition.

Proof Suppose we are given a transfinite sequence Aii∈α indexed

by an ordinal α, with continuity at limit ordinals < α. Put Aα :=

colimi<αAi. Suppose that the transition maps Ai → Ai+1 are global

weak equivalences; and suppose either

(i)—that the transition maps are injective cofibrations, or else

(ii)—that weak equivalences in M are closed under transfinite compo-

sition.

We want to prove that A0 → Aα is a global weak equivalence. By

induction on α we may assume that this statement is known for all

sequences indexed by strictly smaller ordinals, in particular we get that

the transition maps Ai → Aj are global weak equivalences for all i <

j < α.

Set Xi := Ob(Ai) and X := Ob(Aα). Then X is the filtered colimit

of the Xi. Operating by induction on i, we will choose a sequence of


morphisms Ai → A′i where A′

i ∈ PC(Xi,M ) is a weak equivalent

replacement of Ai in the injective model structure PCinj(Xi,M ) for

each i, such that A′i satisfies the Segal conditions and such that these

are compatible in the sense that for i < j < α we have a commutative

diagram

Ai → Aj

A′i

↓

→ A′j .

↓

To make this choice, suppose it is done for all i < j.

If j is a limit ordinal, set A′j := colimi<jA′

i. By an argument which we

will use below for the colimit at α (which we don’t repeat here because

the notations will be more comfortable later), the map Aj → A′j is a

weak equivalence in PC(Xj ,M ) and A′j satisfies the Segal conditions.

This treats the case of a limit ordinal.

Suppose j = i+ 1 is a successor ordinal. Consider the diagram

Ai → Aj

A′i

↓

→ Pj

↓

where Pj := A′i ∪

Ai Aj is defined as the pushout. Now, the left ver-

tical map is a weak equivalence, by the inductive hypothesis on the

statement we are trying to prove, and it is also an injective cofibration

in PC(Xi,M ). By Lemma 16.3.3, pushout along a trivial cofibration

which induces an isomorphism on sets of objects, is again a trivial cofi-

bration. Thus Aj → Pj is a trivial cofibration, and we can let Pj → A′j

be a K-injective replacement of Pj , which will also be a K-injective re-

placement of Aj accepting compatible maps from all the A′i for i < j.

Here K denotes the pseudo-generating set for PC(Xj ,M ) considered in

Chapter 14.

This completes the choice of the sequence A′i, which by construction is

continuous at limit ordinals. Put A′α := colimi<αA′

i. As promised above,

we show that A′α satisfies the Segal conditions and Aα → A′

α is a weak

equivalence in PC(X,M ).

Suppose given a sequence of objects x0, . . . , xn ∈ X . It comes from


a sequence xk0 , . . . , xkn ∈ Xk for some k < α (which depends on the

sequence), and denote by xi0, . . . , xin ∈ Xi the image sequence for any

i ≥ k. The colimit morphism objects for A′α are

A′α(x0, . . . , xn) = colimk≤i<αA

′i(x

i0, . . . , x

in).

Commutation of direct products with colimits, condition (DCL) on M ,

now tells us that

A′α(x0, x1)× · · · × A

′α(xn−1, xn)

= colimk≤i<αA′i(x

i0, x

i1)× · · · × A

′i(x

in−1, x

in).

Transfinite composition of a sequence of weak equivalences is again a

weak equivalence, so the Segal maps for A′α are weak equivalences. Thus,

A′α satisfies the Segal conditions.

By the construction above, the mapAα → A′α is obtained by a transfi-

nite composition of pushouts along pseudo-generating trivial cofibrations

for the various PC(Xi,M ), thus it is a trivial cofibration.

We can now show that A0 → Aα is a global weak equivalence. Use

Aα → A′α as replacement satisfying the Segal conditions, so the problem

is to show that A′0 → A

′α is a weak equivalence.

It is essentially surjective, indeed given any x ∈ X = Ob(A′α) there

is some i < α such that x comes from xi ∈ Xi. Hence the isomorphism

class of x in the truncation, is in the image of

Isoτ≤1(A′i) → Isoτ≤1(A

′α).

However, since A′0 → A

′i is a global weak equivalence, it is essentially

surjective in other words

Isoτ≤1(A′0) → Isoτ≤1(A

′i).

is surjective. It follows that the isomorphism class of x is in the image

of

Isoτ≤1(A′0) → Isoτ≤1(A

′α).

This shows essential surjectivity.

To show that the map is fully faithful, let x0, . . . , xn be a sequence

of objects in X0 = Ob(A′0). Let x

ij denote their images in Xi including

i = α. Now

A′α(x

α0 , . . . , x

αn) = colimi<αA

′i(x

i0, . . . , x

in).


To show full faithfulness we need to show that

A′0(x0, . . . , xn) → colimi<αA

′i(x

i0, . . . , x

in) (16.3.1)


Recall that we are assuming either (i) or (ii) above. In case (i), the

transition maps Ai → Aj are injective cofibrations, and by construction

ofA′i the same is true of the transition maps A′

i → A′j . Hence the colimit

expression above is a sequential colimit whose transition maps are trivial

cofibrations, thus the map (16.3.1) is a trivial cofibration, which shows

that A′0 → A

′α is fully faithful.

In case (ii), we just know that the transition maps in the colimit

are weak equivalences, but the hypothesis (ii) says again that the map

(16.3.1) is a weak equivalence.

In either case, A′0 → A

′α is fully faithful.

16.4 An elementary generation step Gen

Corollary 16.2.5 looks at the replacement A → Seg(A) from the point

of view of direct left Bousfield localizing systems of Chapters 13 and 14.

It will also be useful to have an approach which is more homotopically

canonical, in other words some kind of process which looks canonical

when viewed in the homotopy category ho(Func(∆oX/X,M )) of unital

diagrams up to levelwise weak equivalences. The full process will be

broken up into elementary steps denoted A → Gen(A, q). These should

be thought of as “calculating generators and relations at the Segal map

corresponding to q”.

Fix a set of objects X . Recall that the underlying category used is Φ =

∆oX ; the subcategory on which the unitality condition will be imposed

is Φ0 = (x0)x0∈X , and the set Q used to determine the algebraic

theory of Segal categories over X , is just the object set of ∆X that is

the set of sequences (x0, . . . , xn) of xi ∈ X . For q = (x0, . . . , xn), we have

nq := n; the functor Pq : ǫ(n) → ∆oX sends ξ0 to q = (x0, . . . , xn) and for

1 ≤ j ≤ n it sends ξj to the adjacent pair (xj−1, xj). The structural maps

for Pq come from inclusions of each adjacent pair in the full sequence;

they are the maps which together make up the Segal maps.

An A ∈ PC(X ;M ) is a functor A : ∆oX →M such that A(x) = ∗

for any single element sequence (x).

For any q = (x0, . . . , xn) ∈ Q = Ob(∆oX), the functor

U!Pq,! : Func(ǫ(n),M ) → PC(X,M )


sends injective cofibrations to Reedy cofibrations. Recall that it sends

the standard ρn(f) to the cofibrations Ψ([n], f) see Corollary 16.2.5.

Abbreviate the right adjoint P ∗q U

∗ by just P ∗q .

For A ∈ PC(X,M ) and q = (x0, . . . , xn) we have considered in Sec-

tion 13.6 the cofibration A → Gen(A, q) which depends on a choice of

factorization

P ∗q (A)0 → E → P ∗

q (A)1 × · · · × P∗q (A)n.

Note that P ∗q (A)0 = A(x0, . . . , xn) while P ∗

q (A)j = A(xj−1, xj) for 1 ≤

j ≤ n. Thus, the factorization we need to choose may be written as

A(x0, . . . , xn)e→ E

(p1,...,pn)→ A(x0, x1)× · · · × A(xn−1, xn).

Given such a factorization, which will generally be chosen so that e is a

cofibration and (p1, . . . , pn) is a weak equivalence, we get a map

gen(A, q) : A → Gen(A, q)[E, e, p1, . . . , pn].

If no confusion arises, we abbreviate the right side to Gen(A, q). In the

notation of the previous section,

Gen(A, q)[E, e, p1, . . . , pn] = A ∪srcΨ([n],e) h([n], E). (16.4.1)

Lemma 16.4.1 Let A ∈ PC(X,M ) and q = (x0, . . . , xn), and sup-

pose given a choice of factorization E, e, p1, . . . , pn as above. Suppose e

is a cofibration and (p1, . . . , pn) : E → A(x0, x1)× · · · × A(xn−1, xn) is

a weak equivalence in M . Then gen(A, q) is a trivial cofibration from

A to Gen(A, q) = Gen(A, q)[E, e, p1, . . . , pn] in the model structure

PCReedy(X ;M ) constructed in Theorem 14.3.2.

Proof Indeed gen(A, q) is a pushout along Ψ([n], e) by (16.4.1).

We can describe explicitly the structure of Gen(A;x0, . . . , xn) de-

pending on the choice of E, e, p1, . . . , pn. For any sequence (z0, . . . , zm),

let

∆NSX ((z0, . . . , zm), (x0, . . . , xn)) ⊂ ∆X((z0, . . . , zm), (x0, . . . , xn))

be the subset of maps φ : (z0, . . . , zm) → (x0, . . . , xm) in ∆X which don’t

factor through any of the adjacent pairs (xj−1, xj). Abbreviate this by

∆NSX (z·, x·) when convenient. Then Gen(A;x0, . . . , xn)(z0, . . . , zm) is a


pushout in the diagram∐

φ∈∆NSX (z·,x·)

A(x0, . . . , xn) →∐

φ∈∆NSX (z·,x·)

E

A(z0, . . . , zm)

↓

→ Gen(A;x0, . . . , xn)(z0, . . . , zm).

↓

In order to define the functoriality it is convenient to rewrite this as fol-

lows: for any object B ∈M and any φ ∈ ∆X((z0, . . . , zm), (x0, . . . , xn))

denote by alt(φ,B) either: B if φ ∈ ∆NSX (z·, x·); or ∗ if φ factors through

a singleton

(z0, . . . , zm) → (xj) → (x0, . . . , xn);

or A(xj−1, xj) if φ factors through a map

(z0, . . . , zm) → (xj−1, xj) → (x0, . . . , xn)

but not through a singleton (in which case the choice of j is unique).

Now, for either B = A(x0, . . . , xn) or B = E, we have structural maps

B → A(xj−1, xj). Using these, given a map ψ : (y0, . . . , yp) → (z0, . . . , zm)

then for any φ ∈ ∆X((z0, . . . , zm), (x0, . . . , xn)) we get a map

alt(φ,B) → alt(φψ,B)

and these are compatible with composition in ψ. Indeed, if φ factors

through an adjacent pair or a singleton, then so does φψ. If φψ factors

through a singleton then the map from alt(φ,B) is the unique map to

alt(φψ,B) = ∗. If φψ factors through an adjacent pair but φ didn’t

factor through an adjacent pair, then the map

alt(φ,B) = A(x0, . . . , xn) or E → alt(φψ,B) = A(xj−1, xj)

is the given structural map.

Now with these notations, Gen(A;x0, . . . , xn)(z0, . . . , zm) can also be

expressed as a pushout of the form∐

φ∈∆X(z·,x·)

alt(φ,A(x0, . . . , xn)) →∐

φ∈∆X(z·,x·)

alt(φ,E)

A(z0, . . . , zm)

↓

→ Gen(A;x0, . . . , xn)(z0, . . . , zm).

↓

Note that for a map φ : z· → x· factoring through (xj−1, xj) (resp. (xj))


we get a map (z0, . . . , zm) → (xj−1, xj) (resp. to (xj)) and hence a map

A(xj−1, xj) → A(z0, . . . , zm) (resp. a map ∗ = A(xj) → A(z0, . . . , zm)).

These combine to give the left vertical map.

Now, in case of a map ψ : (y0, . . . , yp) → (z0, . . . , zm) we get maps∐

φ∈∆X(z·,x·)

alt(φ,A(x0, . . . , xn)) →∐

φ∈∆X(z·,x·)

alt(φψ,A(x0, . . . , xn))

→∐

ζ∈∆X(y·,x·)

alt(ζ,A(x0, . . . , xn))

and∐

φ∈∆X(z·,x·)

alt(φ,E) →∐

φ∈∆X(z·,x·)

alt(φψ,E) →∐

ζ∈∆X(y·,x·)

alt(ζ, E)

which induce the map of functoriality

Gen(A;x0, . . . , xn)(z0, . . . , zm) → Gen(A;x0, . . . , xn)(y0, . . . , yp).

This whole discussion can be simplified considerably in the case when

the x0, . . . , xn are all distinct in other words ∀0 ≤ i 6= j ≤ n, xi 6= xj .

Then, for any sequence of objects (z0, . . . , zm) ∈ ∆X there is at most

one map φ : (z0, . . . , zm) → (x0, . . . , xn). It factors through an adjacent

pair if and only if z· is of the form (xj−1, . . . , xj−1, xj , . . . , xj) and it

factors through a singleton if and only if z· = (xj , . . . , xj). There is a

map, but not factoring through an adjacent pair or a singleton, if and

only if

z· = (xi0 , . . . , xim)

where 0 ≤ i0 ≤ i1 ≤ · · · ≤ im ≤ n and i0 + 1 < im. We state the result

in this case as a lemma.

Lemma 16.4.2 Suppose A ∈ PC(X,M ) and (x0, . . . , xn) is a se-

quence of pairwise disjoint objects. Suppose given a choice of factoriza-

tion E, e, p1, . . . , pn as above. Then for any increasing sequence of indices

0 ≤ i0 ≤ i1 ≤ · · · ≤ im ≤ n with i0 + 1 < im,

Gen(A;x0, . . . , xn)(xi0 , . . . , xim) = A(xi0 , . . . , xim ) ∪A(x0,...,xn) E.

For any other sequence of objects (z0, . . . , zm) we have

Gen(A;x0, . . . , xn)(z0, . . . , zm) = A(z0, . . . , zm).

These expressions are compatible with the natural maps from A(z0, . . . , zm),

and the maps of functoriality are given by the structural maps for E in

16.5 Fixing the fibrant condition locally 347

the case of a map ψ : (z0, . . . , zm) → (xi0 , . . . , xim) which factors through

a sequence projecting to an adjacent pair in (x0, . . . , xn). The maps of

functoriality are given by the degeneracy maps of A in case ψ factors

through a repeated singleton.


16.5 Fixing the fibrant condition locally

Applying the operation Gen may destroy the fibrancy condition in the

plain diagram Reedy structure. Let fix : A → Fix(A) be a replacement

by a fibrant object in the Reedy model structure on the plain diagram

category FuncReedy(∆oX/X,M ) where X = Ob(A). The map fix(A) is

a pushout along elements of KReedy namely the generators for the trivial

cofibrations of FuncReedy(∆oX/X,M ). In particular it is an isotrivial

cofibration and levelwise a trivial cofibration. This doesn’t change the

homotopy type in ho(Func(∆oX/X,M )).

16.6 Combining generation steps

We can put together the elementary generation steps defined above, to

obtain a model for the replacement to a Segal object.

Theorem 16.6.1 Suppose A ∈ PC(X,M ). There is a transfinite com-

position sequence Aii∈[α] indexed by an ordinal [α] = α + 1, with

A0 = A, such that A· is continuous at limit ordinals, such that Aα sat-

isfies the Segal conditions, and such that each Ai → Ai+1 has the form

fixgen(Ai, qi) for some qi = (xi,0, . . . , xi,ni). In other words, Ai+1 =

FixGen(Ai, qi)[Ei, ei, pi,1, . . . , pi,ni ] as considered above. Furthermore

the transition maps are all trivial cofibrations in PCReedy(X ;M ), in

particular A → Aα is a trivial cofibration. Denote the end result by

SegFG(A); it is a fibrant object in the Reedy model structure PC(X,M )

and equivalent to Seg(A) in the Reedy model structure on the plain dia-

gram category FuncReedy(∆oX/X,M ), indeed it is a possible choice for

the fibrant replacement Seg(A).

Proof The gen(Ai, qi) are trivial cofibrations which may be chosen to

contain a pushout along any particular cofibration of the form Ψ([n], f).

Similarly the maps fix may be chosen to contain any of the pushouts


along generating trivial cofibrations for the level structure. Thus, the

maps envisioned above may contain all pushouts along elements of the

pseudo-generating set KReedy. Choose the sequence using the small ob-

ject argument so that the end result Aα is in inj(KReedy). In particular

it satisfies the Segal conditions. Note that KReedy-injective objects are

also fibrant, see Theorem 14.3.2 where that statement really comes from

Theorem 11.7.1. Thus, A → SegFG(A) is a trivial cofibration towards

a fibrant object in PCReedy(X,M ).

An important corollary of this procedure is the following statement,

tautological for maps between Segal categories.

Corollary 16.6.2 Suppose a map f : A → B in PC(M ) induces a

weak equivalence

A(x0, . . . , xp)∼→ B(f(x0), . . . , f(xp))

for any sequence (x0, . . . , xp) ∈ ∆Ob(A). Then f is fully faithful.

Proof In the process of Theorem 16.6.1 this condition is preserved at

each step. It follows that SegFG(A) → SegFG(B) is fully faithful, which

is by definition the full faithfulness criterion for f .

16.7 Functoriality of the generation process

Suppose M and N are tractable left proper cartesian model categories.

Suppose F : M → N is a left Quillen functor. We say that F is

compatible with products if, for any A,B ∈ M the natural map F (A ×

B) → F (A) × F (B) is a weak equivalence in N .

The functor F induces functorsPC(X,F ) : PC(X,M ) → PC(X,N )

and hence, asX varies, it induces a functorPC(F ) : PC(M ) → PC(N ).

For A ∈ PC(X,M ) and a sequence of objects x0, . . . , xn in X ,

F (A)(x0, . . . , xn) := F (A(x0, . . . , xn)).

Lemma 16.7.1 In the above situation, suppose A ∈ PC(X,M ), choose

q = (x0, . . . , xn), and suppose given a choice of factorization E, e, p1, . . . , pnas above. Suppose e is a cofibration and (p1, . . . , pn) : E → A(x0, x1)×

· · ·×A(xn−1, xn) is a weak equivalence in M . Then F (e) is a cofibration

and

(Fp1, . . . , Fpn) : F (E) → F (A)(x0, x1)× · · · × F (A)(xn−1, xn)

16.7 Functoriality of the generation process 349

is a weak equivalence in N , so (F (E), F (e), F (p1), . . . , F (pn)) constitute

data for defining

F (A)gen(F (A),q)

→ Gen(F (A), q) = Gen(F (A), q)[F (E), F (e), F (p1), . . . , F (pn)]

in PC(X,N ). In these terms we have gen(F (A), q) = F (gen(A, q))

and

Gen(F (A), q)[F (E), F (e), F (p1), . . . , F (pn)] = F (Gen(A, q)[E, e, p1, . . . , pn])

written more succinctly as Gen(F (A), q) = F (Gen(A, q)).

Proof By inspection.

Corollary 16.7.2 In our above situation of a left Quillen functor

F : M → N compatible with products, between two tractable left proper

cartesian model categories, suppose A ∈ PC(X,M ) and let Aii∈[α] be

a transfinite composition of elementary generation steps in PC(X,M )

with A = A0 → Aα a trivial cofibration in PCinj(X,M ) and Aα satisfy-

ing the Segal conditions. Then F (Ai)i∈[α] is a transfinite composition

of elementary generation steps in PC(X,N ), the map F (A) → F (Aα)

is a trivial cofibration in PCinj(X,N ), and F (Aα) satisfies the Segal

conditions as an N -enriched precategory.

Proof Just apply F to the sequence of Theorem 16.6.1.

We state the next corollary as a theorem, since it is the main statement

we need from this section. Recall from Proposition 14.7.2 that given a

left Quillen functor between tractable left proper cartesian model cate-

gories, it induces a left Quillen functor on the Reedy model categories of

precategories. One didn’t even need to suppose that F preserves prod-

ucts. If that is the case, the statement can be slightly improved to say

that PC(X,F ) preserves weak equivalences (rather than just trivial cofi-

brations).

Proposition 16.7.3 Suppose F : M → N is a left Quillen func-

tor, preserving weak equivalences and compatible with products, between

two tractable left proper cartesian model categories. Then PC(X ;F ) :

PCReedy(X,M ) → PCReedy(X,N ) is a left Quillen functor which

takes weak equivalences to weak equivalences.

Proof Suppose A → B is a weak equivalence in PCReedy(X,M ). This

means that SegFG(A) → Seg

FG(B) is a levelwise weak equivalence

of diagrams, so applying PC(X,F ) gives a new levelwise weak equiv-

alence of diagrams towards N . By the previous corollary, when we


apply PC(X,F ) we get SegFG(PC(X,F )A) → SegFG(PC(X,F )B),

it follows that PC(X,F )A) → PC(X,F )B is a weak equivalence in

PCReedy(X,N ).

16.8 Example: generators and relations for

1-categories

It is interesting and instructive to consider the case where M = Set

is the model category of sets. Here, the weak equivalences are isomor-

phisms, and the fibrations and cofibrations are arbitrary maps. It is

easily seen to be tractable, left proper and cartesian. The category of

Set-enriched precategories PC(Set) may be identified with the cate-

gory of simplicial sets. For A ∈ PC(Set), if we make this identification

then the 0-simplices correspond to the objects; the 1-simplices give gen-

erators for the morphisms, and the 2-simplices give relations among the

morphisms. The Segal conditions for A ∈ PC(Set) are exactly the clas-

sical conditions which are equivalent to stating that A is the nerve of

a category. The calculus of generators and relations constructed above,

reduces in this case to the classical calculus of generators and relations

for 1-categories.

Fix a set X . For this section, the main part of the data of A ∈

PC(X,Set) consists of sets A(x, y) for each pair x, y ∈ X , and A(x, y, z)

for each triple x, y, z ∈ Z. Recall that A(x) is a singleton, so the de-

generacy maps yield elements denoted 1x ∈ A(x, x). Similarly, for any

f ∈ A(x, y) the degeneracies yield elements denoted [1yf ] ∈ A(x, y, y)

and [f1x] ∈ A(x, x, y). In the case f = 1x there is no confusion as both

elements denoted [1x1x] correspond to the same element of A(x, x, x)

obtained from the singleton A(x) by degeneracy. We have projections

i∗01 : A(x, y, z) → A(x, y), i∗12 : A(x, y, z) → A(y, z),

i∗02 : A(x, y, z) → A(x, z).

These are compatible with the degeneracies in an obvious way, for ex-

ample i∗02[1yf ] = i∗02[f1x] = f .

The elements of A(x, y) are the “generating arrows” between x and

y. An element r ∈ A(x, y, z) corresponds to an “elementary relation” of

the form f = gh, where

f = i∗02(r) ∈ A(x, z), g = i∗12(r) ∈ A(y, z), h = i∗01(r) ∈ A(x, y).

16.8 Example: generators and relations for 1-categories 351

This can be seen by looking more closely at what happens under the

generation step Gen(A, q) for a triple q = (x, y, z). Such a step involves

a choice of factorization

A(x, y, z)e→ E

(p1,p2)→ A(x, y)×A(y, z).

All maps are cofibrations in Set so there is no restriction on e; on the

other hand the weak equivalences are the isomorphisms, so (p1, p2) has to

be an isomorphism and we may in effect suppose E = A(x, y)×A(y, z).

We can describe explicitly the resulting precategory A′ := Gen(A, q).

Suppose (u, v) is a pair of elements of X . We need to consider the maps

φ : (u, v) → (x, y, z) in ∆X . There are 6 possibilities:

(u, v) = (x, y)i01→ (x, y, z),

(u, v) = (y, z)i12→ (x, y, z),

(u, v) = (x, z)i02→ (x, y, z),

(u, v) = (x, x)i00→ (x, y, z),

(u, v) = (y, y)i11→ (x, y, z),

(u, v) = (z, z)i22→ (x, y, z).

In case of coincidences among the x, y, z these can overlap in the sense

that the same (u, v) could have several maps to (x, y, z). However, in

the notation of Section 16.4, the only one of these maps which is in

∆NSX ((u, v), (x, y, z)) is i02. Thus, using E = A(x, y) × A(y, z) we have

a pushout diagram

A(x, y, z) → A(x, y)×A(y, z)

A(x, z)

↓

→ A′(x, z).

↓

In other words, A′(x, z) is obtained by taking the old A(x, z) and adding

the symbols gh for g ∈ A(y, z) and h ∈ A(x, y), subject to the relations

that f = gh whenever there is an element of A(x, y, z) mapping to

f ∈ A(x, z), h ∈ A(x, y) and g ∈ A(y, z).

Similarly, if (u, v, w) is a triple of elements, the only maps

(u, v, w) → (x, y, z)


which are in ∆NSX ((u, v), (x, y, z)) are the identity (x, y, z) → (x, y, z),

and the degeneracies of the previous map i02

(x, z, z) → (x, y, z), (x, x, z) → (x, y, z).

The pushout expression along the identity says that for any such symbol

gh which is added to A′(x, z) following the previous discussion, there

will also be a corresponding element of A′(x, y, z) saying that it is the

composition of g and h, in other words the formal composition gh will

be recorded as a composition of g and h. The degeneracies of i02 take

care of keeping track of left and right identities for the new morphisms

gh which are added.

We leave as an exercise to verify that the generation steps Gen(A, q)

at quadruples q = (w, x, y, z) correspond to enforcing the associativity

axiom.

Putting this all together, we see that after repeating the generation

steps infinitely many times for each uplet of objects, the resulting pre-

category SegFG(A) is the nerve of a category, and it is the category

generated by the original arrows A(x, y) subject to the original relations

A(x, y, z).

17

Generators and relations for Segal categories

In this chapter, we consider one of the main examples of the theory:

when M = K is the Kan-Quillen model category of simplicial sets.

Theorem 17.0.1 The Kan-Quillen model category K of simplicial

sets, is a tractable left proper cartesian model category.

Proof Cofibrations are just monomorphisms of simplicial sets, so the

cartesian and tractability conditions are easy to check. Condition (PROD)

of Definition 10.0.9 for trivial cofibrations is Eilenberg-Zilber.

Corollary 17.0.2 For a fixed set of objects X, we get model cate-

gories of K -enriched precategories over X with either the projective

structure PCproj(X ;K ), or the Reedy structure PCReedy(X ;K ) which

is the same as the injective structure PCinj(X ;K ).

Proof Apply Theorems 14.1.1 and 14.3.2. Note that K is a presheaf

category and the cofibrations are monomorphisms, so by Proposition

15.7.2 the Reedy and injective cofibrations coincide. The weak equiva-

lences are the same in all structures so the Reedy and injective structures

are the same.

A K -precategory is called a Segal precategory; those satisfying the

Segal conditions are called Segal categories. We have defined the classes

of global weak equivalences, and two flavors of cofibrations (hence fibra-

tions), on the way to constructing the projective and Reedy=injective

model structures on the category PC(K ) of Segal precategories in

the next part. The notations of Section 12.7 apply, so a Segal pre-

category may be considered as a bisimplicial set (m 7→ Am/ ∈ K or

m,n 7→ Am,n ∈ Set, see below) whose simplicial set A0/ in degree 0 is

the discrete set Ob(A).


354 Generators and relations for Segal categories

The theory of Segal categories goes back a long time. They were first

considered by Dwyer, Kan, Smith [92] and Schwanzl and Vogt [184], but

of course the definition goes essentially back to Segal [187] who just re-

stricted to the case when there is only one object. Other references are

[3] [208] [74]. Bergner [34] and Pelissier [171] have already given com-

plete constructions of the global model category structure. The model

categories PCproj(K ) and PCReedy(K ) which we are in the process of

constructing, are the same as those of Bergner [34].

17.1 Segal categories

Recall that if A is a Segal category, then the truncation τ≤1(A) was

defined in Section 14.5. It is the usual 1-category whose nerve is the

simplicial set m 7→ π0(Am/). A Segal groupoid is a Segal category such

that τ≤1(A) is a groupoid.

The category K satisfies the additional Condition 12.7.1 (DISJ) on

disjoint unions. ThereforePC(K ) may be viewed as a presheaf category.

More precisely, as K is the category of presheaves on ∆, by Theorem

12.7.3 we have a natural identification

PC(K ) ∼= SetC(∆) ⊂ Set

∆×∆. (17.1.1)

A Segal precategory for us is a pair (X,A) where X ∈ Set and A is a

collection of simplicial sets A(x0, . . . , xn) for sequences of xi ∈ X . Recall

that C(∆) is a quotient of ∆ ×∆, thus the subset relation in (17.1.1).

Under the identification (17.1.1) this corresponds to a bisimplicial set

given by the formula

A : (n,m) ∈ C(∆) 7→∐

(x0,...,xn)∈Xn+1

A(x0, . . . , xn)m.

If n = 0 then this is constant in m ∈ ∆, which is Tamsamani’s con-

stancy condition in this case. The constancy condition characterizes the

bisimplicial sets which are presheaves on the quotient C(∆).

In what follows, we use the following notation of Section 12.7 for a

Segal precategory A:

—A0 = Ob(A) = A(0,m) for any m;

—An/ is the simplicial set m 7→ A(n,m).

The assignment n 7→ An/ is a functor ∆o → K , which is the same as

the bisimplicial set A. We denote the first component by A0 rather than

17.2 The Poincare-Segal groupoid 355

A0/ to emphasize the constancy condition, that it is a set considered as

a constant simplicial set.

This notation is convenient because colimits (resp. limits) in PC(K )

correspond to levelwise colimits (resp. limits) of the functors ∆o → K .

For a more pictorial point of view, it is often more intuitive to replace

the model category K by the category Top of topological spaces, and

to think of Segal (pre)-categories as functors ∆oX → Top. As usual,

for technical details it is more convenient to use simplicial sets, and we

don’t treat the thorny questions surrounding model category structures

on Top. Below we will sometimes replace K by Top and leave to the

reader to insert the appropriate realization and singular complex func-

tors between these two.

A Segal precategory may therefore be thought of as a functor A :

∆o → Top denoted n 7→ An/, which is to say a simplicial space, sat-

isfying the “contancy” or “globular” condition that A0 = Ob(A) is a

discrete set.

17.2 The Poincare-Segal groupoid

Given a Kan simplicial set X , that is a fibrant object in K , we can

define its Poincare-Segal groupoid ΠS(X) which is a Segal groupoid, i.e.

a Segal category whose τ≤1 is a groupoid. It is constructed as the right

adjoint of the diagonal realization functor, which we consider first.

The diagonal d : ∆ → ∆×∆ provides a pullback functor

d∗ : Set∆×∆ → Set∆ = K ;

so composing with (17.1.1) we obtain the realization functor

| | : PC(K ) → K ,

|(X,A)| := d∗A =(m 7→ Am,m

).

Recall that c∆ : ∆×∆ → C(∆) denotes the projection map. We obtain

a functor

c∆ d : ∆ → C(∆),

and the realization functor is just the pullback

|A| = (c∆ d)∗(A). (17.2.1)


from PC(K ) ∼= SetC(∆) to K = Set

∆. It follows that | | preserves

limits and colimits, indeed it has both left and right adjoints.

Taking the right adjoint of the expression (17.2.1), define

ΠS := (c∆ d)∗ : K → PC(K ).

Note that ΠS commutes with limits. For a fibrant simplicial set X , call

ΠS(X) the Poincare-Segal groupoid of X .

We now state the general theorem relating the homotopy theory in

PC(K ) of Segal groupoids, with the classical homotopy theory of sim-

plicial sets. It is essentially due to Segal, although Segal’s arguments

were mostly stated for the situation of categories with a single object.

These were reviewed in the more general categorical setting by Tam-

samani [206]. Tamsamani then furthermore iterated the result to obtain

an equivalence between the theory of n-truncated homotopy types, and

their Poincare n-groupoids. We don’t delve into the details of this, refer-

ing the reader to [206] instead.

The theorem essentially says that we have a Quillen adjunction, al-

though at the present point in our argument we haven’t yet shown that

the given classes of morphisms provide a model structure for PC(K ).

Of course that statement is also contained in the references [34] [171].

Theorem 17.2.1 The realization functor | | sends injective cofibra-

tions (resp. injective global trivial cofibrations) to cofibrations (resp. triv-

ial cofibrations) of simplicial sets. The Poincare-Segal groupoid functor

sends fibrations in K to new fibrations in PC(K ). If X is a fibrant

simplicial set then ΠS(X) is fibrant, in particular it is a Segal category.

It is, in fact, a Segal groupoid, and furthermore the adjunction map

|ΠS(X)| → X

is a weak equivalence. Conversely, if A ∈ PC(K ) is a Segal groupoid

and |A| → Y is a fibrant replacement in K , then the map obtained by

adjunction

A → ΠS(Y )

is a global equivalence of Segal categories.

Proof We refer to Tamsamani [206] for the interpretation of Segal’s

original results [187] in the context of many-object groupoids. See also

Bergner [34], Dwyer, Kan and Smith [92], Berger [30], Duskin [86], and

others.


Tamsamani defines the homotopy groups of an n-nerve in [206]. Taken

at the first level, we can similarly define the homotopy groups of a Segal

groupoid A at any x ∈ Ob(A) by putting

π0(A) := Isoτ≤1A

and for i ≥ 1,

πi(A, x) := πi−1(A(x, x), 1x)

where 1x : ∗ → A(x, x) is the degeneracy map using A(x) = ∗.

From the Segal groupoid condition it follows that we can define the

relation of homotopy on Ob(A) by saying that x ∼ y if and only if

A(x, y) 6= ∅, and then π0(A) = Ob(A)/ ∼.

Lemma 17.2.2 A morphism f : A → B between Segal groupoids is

a global weak equivalence, if and only if π0(A) → π0(B) is surjective

(resp. an isomorphism), and for each i ≥ 1 and each object x ∈ Ob(A),

the induced maps πi(A, x) → πi(B, f(x)) are isomorphisms.

Proof This follows directly from the definition using the criterion that

a map between simplicial sets is a weak equivalence if and only if it

induces an isomorphism on homotopy groups.

Proposition 17.2.3 If A is a Segal groupoid then π0(A) = π0(|A|)

and for any i ≥ 1 and x ∈ Ob(A), πi(A, x) = πi(|A|, |x|). Similarly if Z

is a simplicial set satisfying Kan’s fibrancy condition then π0(ΠS(Z)) =

π0(Z) and for any vertex z ∈ Z0, πi(ΠS(Z), z) = πi(Z, z).

A morphism f : A → B between Segal groupoids is a global weak

equivalence if and only if the induced map on realizations is a weak

equivalence |A| ∼ |B|.

Proof The first part states some of the essential facts about Segal’s

construction [187], entering into the proof of Theorem 17.2.1, see [206].

The second paragraph follows immediately from the first part together

with the previous lemma.

17.3 The calculus

After this first part, the remainder of the chapter is devoted to following

out the calculus of generators and relations in the case of Segal precat-

egories. In the spirit of Segal’s “delooping machine”, this process gives

a good explanation of how the technical machinery introduced in the


previous chapters works. First is a detailed description of how to go

from a Segal precategory to a Segal category, keeping track of the con-

nectivity properties of the intervening spaces. In Section 17.4, we show

how the process leads, in principle, to a calculation of the loop space

of a space. In Section 17.5 below we will be able to follow along what

happens as one of the first nontrivial homotopy groups π3(S2) appears

out of this process. This chapter constitutes a reworked version of the

preprint [195].

In view of the topological motivation, we concentrate on the case of

Segal groupoids and more specifically the case when there is only one

object and A1/ is connected. In this case, a stronger finiteness property

will hold: to arrange things up to a certain level of connectivity, it will

suffice to do a finite number of elementary generation operations which

we denote here by Arr (or later Arr2). Of course, if A1/ is not connected

then we may be in the presence of a fundamental group and it requires,

in principle, an infinite and even undecideable number of operations to

compute the group. Some remarks on this aspect are given near the end

of the chapter.

17.3.1 Arranging in degree m

Suppose A is a Segal precategory with A0 = ∗. We say that A is (m, k)-

arranged if the Segal map

Am/ → A1/ × . . .×A1/

induces isomorphisms on πi for i < k and a surjection on πk.

Note that for l ≥ k, adding l-cells to Am/ or l+1-cells to A1/ doesn’t

affect this property.

We now define an operation where we try to “arrange”A in degree m.

This operation is inspired by the operation Gen considered in Section

16.4. In the present version it will be called

A 7→ Arr(A,m).

Fix m in what follows. Let C be the mapping cone of the Segal map

Am/ → A1/ × . . .×A1/.

To be precise, as a bisimplicial set

C = (I ×Am/) ∪1×Am/ (A1/ × . . .×A1/),

where I is the standard simplicial interval, and the notation is coproduct


of bisimplicial sets (note also that the globular condition is preserved, so

it is a coproduct of Segal precategories). Note that 1×Am/ denotes the

second endpoint of the interval crossed with Am/. We have morphisms

Am/ ⊂a→ C

b→ A1/ × . . .×A1/,

the morphism a being the inclusion of 0×Am/ into I ×Am/ (thus it

is a cofibration i.e. injection of simplicial sets) and the second morphism

b coming from the projection I ×Am/ → Am/. The second morphism b


We now define Arr(A,m): for any p, let

Arr(A,m)p/ := Ap/ ∪(⋃

Am/)

(⋃

p → m

C

)

be the combined coproduct of Ap/ with several copies of the morphism

a : Am/ → C , one copy for each map p → m not factoring through a

principal edge (see below for further discussion of this condition), these

maps inducing Am/ → Ap/.

We need to define Arr(A,m) as a Segal precategory, i.e. as a bisim-

plicial set. For this we need morphisms of functoriality

Arr(A,m)p/ → Arr(A,m)q/

for any q → p. These are defined as follows. We consider a component

of Arr(A,m)p/ which is a copy of C attached along a map Am/ → Ap/corresponding to p → m which doesn’t factor through a principal edge.

If the composed map q → p → m doesn’t factor through a princi-

pal edge then the component C maps to the corresponding compo-

nent of Arr(A,m)1/ . If the map does factor through a principal edge

q → 1 → m then we obtain a map C → A1/ (the component of the

map b corresponding to this principal edge). Compose with the map

A1/ → Aq/ to obtain a map C → Aq/. Note that if the map further

factors

q → 0 → 1 → m

then the map A1/ → Aq/ factors through the basepoint

A1/ → A0 → Aq/,

and our map on C factors through the basepoint. This factorization

doesn’t depend on choice of principal edge containing the map 0 → m.

One can verify that this prescription defines a functor p 7→ Arr(A,m)p/


from ∆ to simplicial sets. This verification will be a consequence of the

more conceptual description which follows.

Let h(m) denote the simplicial set representing the standardm-simplex;

it is the contravariant functor on ∆ represented by the object m. Let

Σ(m) ⊂ h(m) be the subcomplex which is the union of the principal

edges.

Definition 17.3.1 If X is a simplicial set and B is another simplicial

set denote by X ⊠B the bisimplicial set exterior product, defined by

(X ⊠B)p,q := Xp ×Bq.

If B is any simplicial set then putting h(m) or Σ(m) in the first vari-

able, we obtain an inclusion of bisimplicial sets which we denote

Σ(m)⊠B → h(m)⊠B.

Note that these bisimplicial sets are not Segal precategories because they

don’t satisfy the globular condition (they are not constant over 0 in the

first variable). However, that the morphism of simplicial sets

(Σ(m)⊠B)0/ → (h(m)⊠B)0/

is an isomorphism because Σ contains all of the vertices.

If A is a Segal precategory then a morphism h(m) ⊠ B → A is the

same thing as a morphism B → Am/. Similarly, a morphism

Σ(m)⊠B → A

is the same thing as a morphism

B → A1/ ×A0 . . .×A0 A1/.

The morphism of realizations

|Σ(m)⊠B| → |h(m)⊠B|

is a weak equivalence. To see this note that it is the product of |B| and

|Σ(m)| → |h(m)|,

and this last morphism is a weak equivalence (it is the inclusion from

the “spine” of the m-simplex to the m-simplex; both are contractible).

Suppose B′ ⊂ B is an injection of simplicial sets. Put

U := (Σ(m)⊠B) ∪Σ(m)⊠B′

(h(m)⊠B′) ,


and

V := h(m)⊠B.

We have an injection U → V . If A is a Segal precategory then a map

U → A consists of a commutative diagram

B′ → B

Am/

↓

→ A1/ ×A0 . . .×A0 A1/.

↓

The inclusion

Σ(m)⊠B → U

induces a weak equivalence of realizations, because of the fact that the in-

clusion Σ(m)⊠B′ → h(m)⊠B′ does. Therefore the morphism |U| → |V|


We can now interpret our operation Arr(A,m) in these terms. Ap-

plying the previous paragraph to the inclusion Am/ → C, we obtain an

inclusion of bisimplicial sets U → V . We get a map U → A correspond-

ing to the diagram

Am/ → C

Am/

↓→ A1/ ×A0 . . .×A0 A1/.

↓

The left vertical arrow is the identity map, the top arrow is a and the

right vertical arrow is b. The bottom arrow is the Segal map.

It is easy to see that

Arr(A,m) = A∪U V .

In passing, this proves associativity of the previous formulas for functo-

riality of Arr(A,m).

We get

|Arr(A,m)| = |A| ∪|U| |V|.

Since |U| → |V| is a weak equivalence, this implies the


Lemma 17.3.2 The morphism induced by the above inclusion on re-

alizations,

|A| → |Arr(A,m)|


The key observation is the following proposition.

Proposition 17.3.3 Suppose A is a Segal precategory with A0 = ∗ and

A1/ connected. Suppose that A is (m, k−1)-arranged and (p, k)-arranged

for some p 6= m. Then Arr(A,m) is (p, k)-arranged and (m, k)-arranged.

Proof Keep the hypotheses of the proposition. Let C be the cone oc-

curing in the construction Arr(A,m). Denote B := Arr(A,m). Then

the map

a : Am/ → C

is weakly equivalent to a map obtained by adding cells of dimension

≥ k to Am/. This is by the condition that A is (m, k− 1)-arranged. Let

h1, . . . , hu be the k-cells that are attached to Am/ to give C.

We first show that B is (p, k)-arranged. Note that Bp/ is obtained from

Ap/ by attaching a certain number of k-cells, hp → mi for i = 1, . . . , u

indexed by the maps p → m not factoring through the principal edges

of m; plus some cells of dimension ≥ k+1. The higher-dimensional cells

don’t have any effect on the question of whether A is (p, k)-arranged.

On the other hand, B1/ is obtained fromA1/ by attaching cells h1 → mi

for i = 1, . . . , u and indexed by the maps 1 → m not factoring through

the principal edges. Note here that these maps cannot be degenerate,

thus they are the non-principal edges.

Now B1/× . . .×B1/ (product of p-copies) is obtained from A1/× . . .×

A1/ by adding k-cells indexed as ν1 → p(h1 → mi ) where the indexing

1 → p are principal edges and 1 → m are non-principal edges. Then by

adding cells of dimension ≥ k + 1 which have no effect on the question.

The notation ν1 → p refers to the map

A1/ → A1/ × . . .×A1/

putting the base point (i.e. the degeneracy of the unique point in A0) in

all of the factors except the one corresponding to the map 1 → p.

For every principal edge 1 → p there is a unique degeneracy p → 1

inducing an isomorphism 1 → 1 and this establishes a bijection between


principal edges and degeneracies. Thus we may rewrite our indexing of

the k-cells attached to the product above as νp → 1(h1 → mi ).

Now for every pair (p → 1, 1 → m) the composition p → m is a de-

generate morphism, not factoring through a principal edge; and these de-

generate morphisms are all different for different pairs (p → 1, 1 → m).

Thus Bp/ contains a k-cell hp → mi for each i = 1, . . . , u and each of

these maps p → m (plus possibly other cells for other maps p → m but

we don’t use these). Take such a cell hp → mi , and look at its image in

B1/ × . . .× B1/ by the Segal map. The projection to any factor 1 → p

other than the one which splits the degeneracy p → 1, is totally degen-

erate coming from a factorization 1 → 0 → 1, hence goes to the unique

basepoint. The projection to the unique factor which splits the degener-

acy is just the cell h1 → mi . Thus the projection of our cell hp → m

i to

the product is exactly the cell νp → 1(h1 → mi ). This shows that all of

the new k-cells which have been added to the product, are lifted as new

k-cells in Bp/. Together with the fact that Ap/ → A1/ × . . .×A1/ was

an isomorphism on πi for i < k and a surjection for i = k, we obtain the

same property for Bp/ → B1/ × . . .× B1/. Note that the further k-cells

which are attached to Bp/ by morphisms p → m other than those we

have considered above, don’t affect this property. (In general, attaching

k-cells to the domain of a map doesn’t affect this property, but attaching

cells to the range can affect it, which was why we had to look carefully

at the cells attached to B1/). This completes the proof that B remains

(p, k)-arranged.

We now prove that B becomes (m, k)-arranged. Note that Bm/ is ob-

tained by first adding on C to Am/ via the identity map m → m; then

adding some other stuff which we treat in a minute. The Segal map for

B maps this copy of C directly into A1/ × . . .A1/. The fact that C is a

mapping cone for the Segal map means that the map

C → A1/ × . . .A1/

is a homotopy equivalence. In particular, it is bijective on πi for i < k

and surjective for i = k.

Now Bm/ is obtained from C by adding various cells to C along degen-

erate maps m → m. The new k-cells which are added to B1/× . . .×B1/(m-factors this time) are lifted to cells in Bm/ added to C via the degen-

eracies m → m which factor through a principal edge. The argument is

the same as above and we don’t repeat it. We obtain that B is (m, k)-

arranged.

This completes the proof of the proposition.


17.3.2 Connectivity properties of (m,k)-arranged

precategories

Suppose A is a Segal precategory with A0 = ∗ and A1/ connected, such

that A is (m, k)-arranged for all m+ k ≤ n. We would like to obtain a

map A → A′ to a Segal category, which induces an equivalence in the

domain covered by the (m, k)-arrangement conditions which are already

known.

Let Λ(m) ⊂ h(m) denote some “horn”, i.e. a union of all faces but

one. Use the notation of Definition 17.3.1. For any inclusion of simplicial

sets B′ ⊂ B we can set

U := (Λ(m)⊠ B) ∪Λ(m)⊠B′

(h(m)⊠B′)

and

V := h(m)⊠B,

we obtain an inclusion of bisimplicial sets U ⊂ V such that |U| → |V| is

a weak equivalence. If A is a Segal precategory and U → A a morphism

then set A′ := A ∪U V . The morphism |A| → |A′| is a weak equiva-

lence. Note that A′ is again a Segal precategory because the morphism

Λ(m) → h(m) includes all of the vertices of h(m). Finally, note that for

p ≤ m− 2 the morphism

Ap/ → A′p/

is an isomorphism (because the same is true of Λ(m) → h(m)). This last

property allows us to conserve the homotopy type of the smaller Ap/.

We would like to use operations of the above form, to (m, k)-arrange

A. In order to do this we analyze what a morphism from U to A means.

For any simplicial set X , we can form the simplicial set [X,A] with the

property that a map B → [X,A] is the same thing as a map X⊠B → A.

If X → Y is an inclusion obtained by adding on anm-simplex h(m) over

a map Z → X where Z ⊂ h(m) is some subset of the boundary, then

[Y,A] = [X,A]×[Z,A] Am/.

In this way, we can reduce [X,A] to a gigantic iterated fiber product of

the various components Ap/.

Lemma 17.3.4 In the above situation, there exists a cofibration A → A′

such that Ap/ → A′p/ is a weak equivalence for all p, and such that for

any cofibration of simplicial sets X ⊂ Y including all of the vertices of


Y , the morphism

[Y,A′] → [X,A′]

is a Kan fibration of simplicial sets.

Proof In order to construct A′, we just “throw in” everything that

is necessary. More precisely, suppose B′ ⊂ B is a trivial fibration of

simplicial sets. A diagram

B′ → B

[Y,A′]

↓

→ [X,A′]

↓

corresponds to a diagram

U → X ⊠B

A′

↓

→ A′

↓

with U = (Y ⊠ B′) ∪X⊠B′

(X ⊠ B). The morphism of bisimplicial sets

U → X ⊠B is a weak equivalence on each vertical column ( )p/. There-

fore we can throw in to A′ the pushout along this morphism, without

changing the weak equivalence type of the Ap/. Note that the new thing

is again a Segal precategory because of the assumption that X ⊂ Y

contains all of the vertices. Keep doing this addition over all possible

diagrams, an infinite number of times, until we get the required Kan

fibration condition to prove the lemma.

Theorem 17.3.5 If A is a Segal precategory with A0 = ∗ and A1/

connected, such that A is (m, k)-arranged for all m + k ≤ n then there

exists a morphism A → A′ such that:

(1) the morphism |A| → |A′| is a weak equivalence;

(2) A′ is a Segal groupoid; and

(3) the map of simplicial sets Am/ → A′m/ induces an isomorphism on

πi for i+m < n.

Proof The answer A′ is the result of a procedure which we now de-

scribe. At each step of the procedure, the construction of the previous

lemma will be applied without necessarily saying so everywhere. Thus


we may always assume that our Segal precategory satisfies the condition

of Lemma 17.3.4.

We have already described above an arranging operationA 7→ Arr(A,m).

We now describe a second arranging operation, under the hypothesis

that A satisfies the conclusion of Lemma 17.3.4. Fix m and fix a horn

Λ(m) ⊂ h(m) (complement of all but one of the faces, and the face that

is left out should be neither the first nor the last face). Let C be the

cone on the map

Am/ = [h(m),A] → [Λ(m),A].

Thus we have a diagram

Am/ → C → [Λ(m),A].

Note that Λ(m) is a gigantic iterated fiber product of various Ap/ for

p < m. This diagram corresponds to a map U → A where

U := (Λ(m)⊠ C) ∪Λ(m)⊠Am/ h(m)⊠Am/.

Letting V := h(m)⊠ C we set

Arr2(A,m) := A ∪U V .

Notice first of all that by the previous discussion, the map

|A| → |Arr2(A,m)|


We have to try to figure out what effect Arr2(A,m) has. We do this

under the following hypothesis on the utilisation of this operation: that

A is (m, k − 1)-arranged, and (p, k)-arranged for all p < m.

The first step is to notice that the fiber product in the expression

of [Λ(m),A] is a homotopy fiber product, because of the condition of

Lemma 17.3.4 which is imposed on A. Furthermore the elements in this

fiber product all satisfy the Segal condition up to k (bijectivity on πi for

i < k and surjectivity for πk). Thus the morphism

[Λ(m),A] → A1/ ×A0 . . .×A0 A1/

(the Segal fiber product for m, on the right) is an isomorphism on πi for

i < k and a surjection for i = k. Thus when we add to Am/ the cone C,

we obtain the condition of being (m, k)-arranged.

By hypothesis,A is (m, k−1)-arranged, in particular the mapAm/ → C

is an isomorphism on πi for i < k − 1 and surjective for πk−1. Thus C

may be viewed as obtained from Am/ by adding on cells of dimension


≥ k. Therefore, for all p the morphisms Up/ → Vp/ are homotopically

obtained by addition of cells of dimension ≥ k.

From the previous paragraph, some extra cells of dimension ≥ k are

added to various Ap/ in the process. This doesn’t spoil the condition of

being (p, k)-arranged wherever it exists. However, the major advantage

of this second operation is that the Ap/ are left unchanged for p ≤ m−2.

This is because all p-faces of the m-simplex are then contained in the

horn Λ(m).

We review the above results. First, the hypotheses on A were:

(a) that A satisfies the lifting condition of Lemma 17.3.4;

(b) that A is (p, k)-arranged for p < m, and (m, k − 1)-arranged. We

then obtain a construction Arr2(A,m) with the following properties:

(1) the map Ap/ → Arr2(A,m)p/ is an isomorphism for p ≤ m− 2;

(2) for any p the map Ap/ → Arr2(A,m)p/ induces an isomorphism

on πi for i < k − 1;

(3) if A is (p, k)-arranged for any p then Arr2(A,m) is also (p, k)-

arranged;

(4) and Arr2(A,m) is (m, k)-arranged.

Remark: at m = 2 the operations Arr(A, 2) and Arr2(A, 2) coincide.

With an infinite series of applications of the construction Arr2(A,m)

and the replacement operation of Lemma 17.3.4 we can prove Theorem

17.3.5. The reader may do this as an exercise or else read the explanation

below.

Take an array of dots, one for each (p, k). Color the dots green if A is

(p, k)-arranged, red otherwise (note that one red dot in a column implies

red dots everywhere above). We do a sequence of operations of the form

of Lemma 17.3.4 (which doesn’t change the homotopy type levelwise)

and then Arr2(A,m). When we do this, change the colors of the dots

appropriately.

Also mark an × at any dot (p, k) such that the πi(Ap/) change for any

i ≤ k− 1. (Keep any × which are marked, from one step to another). If

a dot (p, k) is never marked with a × it means that the πi(Ap/) remain

unchanged for i < k.

We don’t color the dots (1, k) but we still might mark an ×.

Suppose the dot (m, k) is red, the dots (p, k) are green for p < m

and the dot (m, k − 1) is green. Then apply the replacement operation

of Lemma 17.3.4 and the operation Arr2(A,m). This has the following

effects. Any green dot (p, j) for p ≤ m − 2 (and arbitrary j) remains

green. The dot (m− 1, k) remains green. However, the dot (m− 1, k+1)

becomes red. The dot (m, k) becomes green. The dots (m − 1, k) and


(m, k), as well as all (p, k) for p > m, are marked with an ×. The dots

above these are also marked with an × but no other dots are (newly)

marked with an ×.

In the situation of Theorem 17.3.5, we start with green dots at (p, k)

for p + k ≤ n. We may as well assume that the rest of the dots are

colored red. Start with (m, k) = (n + 1, 0) and apply the procedure of

the previous paragraph. The dot (n+1, 0) becomes green, the dot (n, 0)

stays green, and the dots (n, 0), (n + 1, 0), . . . are marked with an ×.

Continue now at (n, 1) and so on. At the end we have made all of the

dots (p, k) with p+ k = n + 1 green, and we will have marked with an

× all of the dots (p, k) with p+ k = n (including the dot (1, n− 1); and

also all of the dots above this line).

We can now iterate the procedure. We successively get green dots on

each of the lines p + k = n + j for j = 1, 2, 3, . . .. Furthermore, no new

dots will be marked with a ×. After taking the union over all of these

iterations, we obtain an A′ which is (p, k)-arranged for all (p, k). Thus

A′ is a Segal category.

Note that the morphism |A| → |A′| is a weak equivalence of spaces.

By looking at which dots are marked with an ×, we find that the

morphisms

Ap/ → A′p/

induce isomorphisms on πi whenever i < n−p. This completes the proof

of the theorem.

17.3.3 Iteration

The following corollary to Theorem 17.3.5 says that in order to calculate

the n-type of Ω|A| we just have to change A by pushouts preserving the

weak equivalence type of |A| in such a way that A is (m, k)-arranged for

all m+ k ≤ n+ 2.

Corollary 17.3.6 Suppose A is a Segal precategory with A0 = ∗ and

A1/ connected, such that A is (m, k)-arranged for all m+ k ≤ n. Then

the natural morphism

|A1/| → Ω|A|

induces an isomorphism on πi for i < n− 1.


Proof: Use Theorem 17.3.5 to obtain a morphism A → A′ with the

properties stated there (which we refer to as (1)–(3)). We have a diagram

|A1/| → Ω|A|

|A′1/|

↓

→ Ω|A′|

↓

.

By property (1) the vertical morphism on the right is a weak equiva-

lence. By property (2) and Theorem 5.3.1 the morphism on the bottom

is a weak equivalence. By property (3) the vertical morphism on the

right induces isomorphisms on πi for i < n− 1. This gives the required

statement.

Corollary 17.3.7 Fix n, and suppose A is a Segal precategory with

A0 = ∗ and A1/ connected. By applying the operations A 7→ Arr(A,m)

for various m, a finite number of times (less than (n+2)2) in a predeter-

mined way, we can effectively get to a morphism of Segal precategories

A → B such that

|A| → |B|

is a weak equivalence of spaces, and such that B is (m, k)-arranged for

all m+ k ≤ n+ 2. Furthermore B0 = ∗ and B1/ is connected.

Proof: By Corollary 17.3.2 any successive application of the opera-

tions A 7→ Arr(A,m) yields a morphism |A| → |B| which is a weak

equivalence of spaces. By Proposition 17.3.3 it suffices, for example, to

successively apply Arr(A, i) for i = 2, 3, . . . , n + 2 and to repeat this

n + 2 times. These operations preserve connectedness of the pieces in

degree 1, so B1/ is connected.

Corollary 17.3.8 Fix n, and suppose A is a Segal precategory with

A0 = ∗ and A1/ connected. Let B be the result of the operations of

Corollary 17.3.7. Then the n-type of the simplicial set B1/ is equivalent

to the n-type of Ω|A|.

Proof: Apply Corollaries 17.3.6 and 17.3.7.


17.4 Computing the loop space

Suppose X is a simplicial set with X0 = X1 = ∗, and with finitely

many nondegenerate simplices. Fix n. We will obtain, by iterating an

operation closely related to the operation gen of Chapter 16, a finite

complex representing the n-type of ΩX .

Let A be X considered as a Segal precategory constant in the second

variable, in other words

Ap,k := Xp.

Apply the arrangement process, iterated as in Corollary 17.3.7.

Corollary 17.4.1 Fix n, and suppose X is a simplicial set with finitely

many nondegenerate simplices, with X0 = X1 = ∗. Let A be X con-

sidered as a Segal precategory. Let B be the result of the operations of

Corollary 17.3.7. Then the n-type of the simplicial set B1/ is equivalent

to the n-type of ΩX.

Proof: An immediate restatement of 17.3.8.

Remark: Any finite region of the Segal precategory B is effectively

computable. In fact it is just an iteration of operations pushout and

mapping cone, arranged in a way which depends on combinatorics of

simplicial sets. Thus the n+ 1-skeleton of the simplicial set B1/ is effec-

tively calculable (in fact, one could bound the number of simplices in

B1/).

Corollary 17.4.2 Fix n, and suppose X is a simplicial set with finitely

many nondegenerate simplices, with X0 = X1 = ∗. Then we can effec-

tively calculate Hi(Ω|X |,Z) for i ≤ n.

Proof: Immediate from above.

In some sense this corollary is the “most effective” part of the present

argument, since we can get at the calculation after a bounded number

of easy steps of the form A 7→ Arr(A,m).

We describe how to use the above description of ΩX inductively to

obtain the πi(X). This seems to be a new algorithm, different from those

of E. Brown [52] and Kan–Curtis [131] [132] [79].

There is an unboundedness to the resulting algorithm, coming essen-

tially from a problem with π1 at each stage. Even though we know in

advance that the π1 is abelian, we would need to know “why” it is abelian

in a precise way in order to specify a strategy for making A1/ connected


at the appropriate place in the loop. In the absence of a particular de-

scription of the proof we are forced to say “search over all proofs” at

this stage. See Subsection 17.4.5 for further discussion.

17.4.1 Getting A1/ to be connected

In the general situation, we have to tackle the problem of computation

of a fundamental group using generators and relations, known to be

undecideable in general. Some sub-cases can still be treated effectively.

The first question is how to arrange A on the level of τ≤1(A).

We define operations Arr0 only(A,m) and Arr1 only(A,m). These con-

sist of doing the operation Arr(A,m) but instead of using the entire

mapping cone C, only adding on 0-cells to Am/ to get a surjection on

π0; or only adding on 1-cells to get an injection on π0. Note in the second

case that we don’t add extra 0-cells. This is an important point, because

if we added further 0-cells every time we added some 1-cells, the process

would never stop.

To define Arr0 only(A,m), use the same construction as for Arr(A,M)

but instead of setting C to be the mapping cone, we put

C′ := Am/ ∪ sk0(A1/ × . . .×A1/.

Here sk0 denotes the 0-skeleton of the simplicial set, and im means the

image under the Segal map. Let C ⊂ C′ be a subset where we choose

only one point for each connected component of the product. With this

C the same construction as previously gives Arr0 only(A,m).

With the subset C ⊂ C′ chosen as above (note that this choice can

effectively be made) the resulting simplicial set

p 7→ π0(Arr0 only(A,m)p/

)

may be described only in terms of the simplicial set

p 7→ π0(Ap/).

That is to say, this operation Arr0 only(A,m) commutes with the oper-

ation of componentwise applying π0. We formalize this as

τ≤1Arr0 only(A,m) = τ≤1Arr

0 only(τ≤1A,m).

To define Arr1 only(A,m), let C be the cone of the map from Am/ to

im(Am/) ∪ sk1(A1/ × . . .×A1/)o

where sk1(A1/× . . .×A1/)o denotes the union of connected components


of the 1-skeleton of the product, which touch im(Am/). In this case, note

that the inclusion

Am/ → C

is 0-connected (all connected components of C contain elements ofAm/).

Using this C we obtain the operation Arr1 only(A,m). It doesn’t intro-

duce any new connected components in the new simplicial sets A′p/, but

may connect together some components which were disjoint in Ap/.

Again, the operation Arr1 only(A,m) commutes with truncation: we

have

τ≤1Arr1 only(A,m) = τ≤1Arr

1 only(τ≤1A,m).

Our goal in this section is to find a sequence of operations which

makes τ≤1(A)1 become trivial (equal to ∗). In view of this, and the above

commutations, we may henceforth work with simplicial sets (which we

denote U = τ≤1A for example) and use the above operations followed

by the truncation τ≤1 as modifications of the simplicial set U . We try

to obtain U1 = ∗. This corresponds to making A1/ connected.

Our operations have the following interpretation. The operation

U 7→ τ≤1Arr0 only(U, 2)

has the effect of formally adding to U1 all binary products of pairs of

elements in U1. (We say that a binary product of u, v ∈ U1 is defined

if there is an element c ∈ U2 with principal edges u and v in U1; the

product is then the image w of the third edge of c).

The operation


has the effect of identifying w and w′ any time both w and w′ are binary

products of the same elements u, v.

The operation


has the effect of introducing, for each triple (u, v, w), the various binary

products one can make (keeping the same order) and giving a formula

(uv)w = u(vw)

for certain of the binary products thus introduced.

It is somewhat unclear whether blindly applying the composed oper-

ation

U 7→ τ≤1Arr1 only(τ≤1Arr

0 only(U, 3), 2)


many times must automatically lead to U1 = ∗ in case the actual fun-

damental group is trivial. This is because in the process of adding the

associativity, we also add in some new binary products; to which asso-

ciativity might then have to be applied in order to get something trivial,

and so on.

If the above doesn’t work, then we may need a slightly revised ver-

sion of the operation Arr0 only(U, 3) where we add in only certain triples

u, v, w. This can be accomplished by choosing a subset of the original C

at each time. Similarly for the Arr0 only(U, 2) for binary products. We

now obtain a situation where we have operations which effect the appro-

priate changes on U corresponding to all of the various possible steps in

an elementary proof that the associative unitary monoid generated by

generators U1 with relations U2, is trivial. Thus if we have an elemen-

tary proof that the associative unitary monoid generated by U1 with

relations U2 is trivial, then we can read off from the steps in the proof,

the necessary sequence of operations to apply to get U1 = ∗. On the

level of A these same steps will result in a new A with A1/ connected.

In our case we are interested in the group completion of the monoid:

we want to obtain the condition of being a Segal groupoid not just a

Segal category. It is possible that the simplicial set X we start with

would yield a monoid which is not a group, when the above operations

are applied. To fix this, we take note of another operation which can be

applied to A which doesn’t affect the weak type of the realization, and

which guarantees that, when the monoid U is generated, it becomes a

group.

Let I be the category with two objects and one morphism 0 → 1,

and let I be the category with two objects and an isomorphism between

them. Consider these as Segal categories (taking their nerve as bisimpli-

cial sets constant in the second variable). Note that |I| and |I| are both

contractible, so the obvious inclusion I → I induces an equivalence of

realizations.

The bisimplicial set I is just that which is represented by (1, 0) ∈

∆×∆. Thus for a Segal precategory A, if f ∈ A1,0 is an object of A1/

(a “morphism” in A) then it corresponds to a morphism I → A. Set

Af := A ∪I I.

Now the morphism f is strictly invertible in the precategory τ≤1(Af ) and

in particular, when we apply the operations described above, the image

of f becomes invertible in the resulting category. If A0 = ∗ (whence

Af0 = ∗ too) then the image of f becomes invertible in the resulting


monoid. Note finally that

|A| → |Af | = |A| ∪|I| |I|

is a weak equivalence. In fact we want to invert all of the 1-morphisms.

Let

A′ := A∪⋃

f I

⋃

f

I

where the union is taken over all f ∈ A1,0. Again |A| → |A′| is a weak

equivalence. Now, when we apply the previous procedure to τ≤1(A′)

giving a category U (a monoid if A had only one object), all morphisms

coming from A1,0 become invertible. Note that the morphisms in A′, i.e.

objects of A′1,0, are either morphisms in A or their newly-added inverses.

Thus all of the morphisms coming from A′1,0 become invertible in the

category U . But it is clear from the operations described above that U

is generated by the morphisms in A′1,0. Therefore U is a groupoid. In

the case of only one object, U becomes a group.

By Segal’s theorem we then have U = π1(|A|). If we know for some

reason that |A| is simply connected, then U is the trivial group. More

precisely, search for a proof that π1 = 1, and when such a proof is

found, apply the corresponding series of operations to τ≤1(A′) to obtain

U = ∗. Applying the operations to A′ upstairs, we obtain a new A′′ with

|A′′| ∼= |A′| ∼= |A| and A′′1/ connected.

Another way of looking at this is to say that every time one needs to

take the inverse of an element in the proof that the group is trivial, add

on a copy of I over the corresponding copy of I.

17.4.2 The case of finite homotopy groups

We first present our algorithm for the case of finite homotopy groups.

Suppose we want to calculate πn(X). We assume known that the πi(X)

are finite for i ≤ n.

Start: Fix n and start with a simplicial set X containing a finite number

of nondegenerate simplices. Suppose we know that π1(X, x) is a given

finite group; record this group, and set Y equal to the corresponding

covering space of X . Thus Y is simply connected. Now contract out a

maximal tree to obtain Z with Z0 = ∗.

Step 1. Let Ap,k := Zp be the corresponding Segal precategory. It has

only one object.


Step 2. Let A′ be the coproduct of A with one copy of the nerve of

the category I (containing two isomorphic objects), for each morphism

I → A (i.e. each point in A1,0).

Step 3. Apply the procedure of Subsection 17.4.1 to obtain a morphism

A′ → A′′ with A′′1/ connected, and inducing a weak equivalence on

realizations. (This step can only be bounded if we have a specific proof

that π1(Y, y) = 1).

Step 4. Apply the procedure of Corollary 17.4.1 and Theorem 17.3.5 to

obtain a morphismA′′ → B inducing a weak equivalence on realizations,

such that B is a Segal groupoid. Note that the n− 1-type of B1/ is effec-

tively calculable (the non-effective parts of the proof of Theorem 17.3.5

served only to prove the properties in question). By Segal’s theorem,

|B1/| ∼ Ω|B| ∼ Ω|Y |,

which in turn is the connected component of Ω|X |. Thus

πn(|X |) = πn−1(|B1/|).

Step 5. Go back to the Start with the new n equal to the old n− 1, and

the new X equal to the simplicial set B1/ above. The new fundamental

group is known to be abelian (since it is π2 of the previous X). Thus

we can calculate the new fundamental group as H1(X) and, under our

hypothesis, it will be finite.

Keep repeating the procedure until we get down to n = 1 and have

recorded the answer.

17.4.3 How to get rid of free abelian groups in π2

In the case where the higher homotopy groups are infinite (i.e. they

contain factors of the form Za) we need to do something to get past

these infinite groups. If we go down to the case where π1 is infinite, then

taking the universal covering no longer results in a finite complex. We

prefer to avoid this by tackling the problem at the level of π2, with a

geometrical argument. Namely, if H2(X,Z) is nonzero then we can take

a class there as giving a line bundle, and take the total space of the

corresponding S1-bundle. This amounts to taking the fiber of a map

X → K(Z, 2). This can be done explicitly and effectively, resulting

again in a calculable finite complex. In the new complex we will have

reduced the rank of H2(X,Z) = π2(X) (we are assuming that X is

simply connected).

The original method of E. Brown [52] for effectively calculating the πi


was basically to do this at all i. The technical problems in [52] are caused

by the fact that one doesn’t have a finite complex representing K(Z, n).

In the case n = 2 we don’t have these technical problems because we

can look at circle fibrations and the circle is a finite complex. For this

section, then, we are in some sense reverting to an easy case of [52] and

not using the Seifert-Van Kampen technique.

Suppose X is a simplicial set with finitely many nondegenerate sim-

plices, and suppose X0 = X1 = ∗. We can calculate H2(X,Z) as the

kernel of the differential

d : ZX′2 → Z

X′3 .

Here X ′i is the set of nondegenerate i-simplices. (Note that a basis of this

kernel can effectively be computed using Gaussian elimination). Pick an

element β of this basis, which is a collection of integers bt for each 2-

simplex (i.e. triangle) t. For each triangle t define an S1-bundle Lt over

t together with trivialization

Lt|∂t ∼= ∂t× S1.

To do this, take Lt = t × S1 but change the trivialization along the

boundary by a bundle automorphism

∂t× S1 → ∂t× S1

obtained from a map ∂t → S1 with winding number bt. Let L(2) be the

S1-bundle over the 2-skeleton of X obtained by glueing together the Ltalong the trivializations over their boundaries. We can do this effectively

and obtain L(2) as a simplicial set with a finite number of nondegenerate

simplices.

The fact that d(β) = 0 means that for a 3-simplex e, the restriction

of L(2) to ∂e (which is topologically an S2) is a trivial S1-bundle. Thus

L(2) extends to an S1-bundle L(3) on the 3-skeleton of X . Furthermore,

it can be extended across any simplices of dimension ≥ 4 because all

S1-bundles on Sk for k ≥ 3, are trivial (H2(Sk,Z) = 0). We obtain

an S1-bundle L on X . By subdividing things appropriately (including

possibly subdividing X) we can assume that L is a simplicial set with

a finite number of nondegenerate simplices. It depends on the choice of

basis element β, so call it L(β).

Let

T = L(β1)×X . . .×X L(βr)

where β1, . . . , βr are our basis elements found above. It is a torus bundle


with fiber (S1)r. The long exact homotopy sequence for the map T → X

gives

πi(T ) = πi(X), i ≥ 3;

and

π2(T ) = ker(π2(X) → Zr).

Note that Zr is the dual ofH2(X,Z) so the kernel π2(T ) is finite. Finally,

π1(T ) = 0 since the map π2(X) → Zr is surjective.

Note that we have a proof that π1(T ) = 0.

17.4.4 The general algorithm

Here is the general situation. Fix n. Suppose X is a simplicial set with

finitely many nondegenerate simplices, with X0 = ∗ and with a proof

that π1(X) = 1. We will calculate πi(X) for i ≤ n.

Step 1. Calculate (by Gaussian elimination) and record π2(X) = H2(X,Z).

Step 2. Apply the operation described in the previous subsection above,

to obtain a new T with πi(T ) = πi(X) for i ≥ 3, with T0 = ∗, with

π1(T ) = 1, and with π2(T ) is finite.

Let A be the Segal precategory corresponding to T .

Step 3. Use the discussion of Section 17.4.1 to obtain a morphism

A → A′ inducing a weak equivalence of realizations, such that A′1/ is

connected. For this step we need a proof that π1(T ) = 1. In the absence

of a specific (finite) proof, search over all proofs.

Step 4. Use Corollary 17.3.7 to replace A′ by a Segal precategory B

with |A′| → |B| a weak equivalence, such that the n − 1-type of B1/ is

equivalent to Ω|B| which in turn is equivalent to Ω|X |. Let Y = B1/ as

a simplicial set.

Note that Y is connected and π1(Y ) is finite, being equal to π2(T ).

We have πi(X) = πi−1(Y ) for 3 ≤ i ≤ n.

Step 5. Choose a universal cover of Y , and mod out by a maximal

tree in the 1-skeleton to obtain a simplicial set Z, with finitely many

nondegenerate simplices, with Z0 = ∗, and with a proof that π1(Z) = 1.

We have πi(X) = πi−1(Z) for 3 ≤ i ≤ n.

Go back to the beginning of the algorithm and plug in (n − 1) and

Z. Keep doing this until, at the step where we calculate π2 of the new

object, we end up having calculated πi(X) as desired.


17.4.5 Proofs of Godement

We pose the following question: how could one obtain, in the process of

applying the above algorithm, an explicit proof that at each stage the

fundamental group (of the universal cover Z in step 5) is trivial? This

could then be plugged into the machinery above to obtain an explicit

strategy, thus we would avoid having to try all possible strategies. To do

this we would need an explicit proof that π1(Y ) is finite in step 4, and

this in turn would be based on a proof that π1(Y ) = π2(T ) as well as a

proof of the Godement property that π2(T ) is abelian.

17.5 Example: π3(S2)

The story behind the preprint [195] was that Ronnie Brown came by

Toulouse for Jean Pradines’ retirement party, and we were discussing

Seifert-Van Kampen. He pointed out that the result of [193] didn’t seem

to lead to any actual calculations. After that, I tried to use that tech-

nique (in its simplified Segal-categoric version) to calculate π3(S2). It

was apparent from this calculation that the process was effective in gen-

eral.

We describe here what happens for calculating π3(S2). We take as

simplicial model a simplicial set with the basepoint as unique 0-cell ∗

and with one nondegenerate simplex e in degree 2. Note that this leads

to many degenerate simplices in degrees ≥ 2 (however there is only one

degenerate simplex which we denote ∗ in degree 1).

We follow out what happens in a language of cell-addition. Thus we

don’t feel required to take the whole cone C at each step of an operation

Arr(A,m); we take any addition of cells to Am/ lifting cells in A1/ ×

. . .×A1/.

We keep the notation A for the result of each operation (since our

discussion is linear, this shouldn’t cause too much confusion).

The first step is to (2, 0)-arrange A. We do this by adding a 1-cell

joining the two 0-cells in A2/, in an operation of type Arr(A, 2). Note

that both 0-cells map to the same point A1/ ×A1/ = ∗. The first result

of this is to add on 1-cells in the Am/ connecting all of the various

degeneracies of e, to the basepoint. Thus the Am/ become connected.

Additionally we get a new 1-cell added onto to A1/ corresponding to

the third face (02). Furthermore, we obtain all images of this cell by

degeneraciesm → 1. Thus we get m circles attached to the pieces which

17.5 Example: π3(S2) 379

became connected in the first part of this operation. Now each Am/ is a

wedge of m circles.

In particular note that A is now (m, 1)-arranged for all m.

The next step is to (2, 2)-arrange A. To do this, note that the Segal

map is

S1 ∨ S1 = A2/ → A1/ ×A1/ = S1 × S1.

To arrange this map we have to add a 2-cell to S1 ∨ S1 with attaching

map the commutator relation. Again, this has the result of adding on

2-cells to all of the Am/ over the pairwise commutators of the loops.

Furthermore, we obtain an extra 2-cell added onto A1/ via the edge

(02). The attaching map here is the commutator of the generator with

itself, so it is homotopically trivial and we have added on a 2-sphere.

(Note in passing that this 2-sphere is what gives rise to the class of

the Hopf map). Again, we obtain the images of this S2 by all of the

degeneracy maps m → 1. Now

A1/ = S1 ∨ S2,

A2/ = (S1 × S1) ∨ S2 ∨ S2,

and in generalA is (m, 2)-arranged for allm. Looking forward to the next

section, we see that adding 3-cells to Am/ for m ≥ 3 in the appropriate

way as described in the proof of 17.3.5, will end up resulting in the

addition of 4-cells (or higher) to A1/ so this no longer affects the 2-type

of A1/. Thus (for the purposes of getting π3(S2)) we may now ignore

the Am/ for m ≥ 3.

The remaining operation is to (2, 3)-arrange A. For this, look at the

Segal map

A2/ = (S1 × S1) ∨ S2 ∨ S2 →

A1/ ×A1/ = (S1 ∨ S2)× (S1 ∨ S2).

Let C be the mapping cone on this map. Then we end up attaching one

copy of C to A1/ along the third edge map A2/ → A1/. This gives the

answer for the 2-type of ΩS2:

τ≤2(ΩS2) = τ≤2

((S1 ∨ S2) ∪(S

1×S1)∨S2∨S2

C).

To calculate π2(ΩS2) we revert to a homological formulation (because

it isn’t easy to “see” the cone C). In homology of degree ≤ 2, the above


Segal map

(S1 × S1) ∨ S2 ∨ S2 → (S1 ∨ S2)× (S1 ∨ S2)

is an isomorphism. Thus the map A2/ → C is an isomorphism on ho-

mology in degrees ≤ 2, and adding in a copy of C along A2/ doesn’t

change the homology. Thus

H2(ΩS2) = H2(S

1 ∨ S2) = Z.

Noting that (as we know from general principles) π1(ΩS2) = Z acts

trivially on π2(ΩS2) and π1 itself has no homology in degree 2, we get

that π2(ΩS2) = H2(ΩS

2) = Z.

Exercise: Calculate π4(S3) using the above method.

Remark: our above recourse to homology calculations suggests that

it might be interesting to do pushouts and the operation Cat in the

context of simplicial chain complexes.

17.5.1 Seeing Kan’s simplicial free groups

Using the above procedure, we can actually see how Kan’s simplicial

free groups arise in the calculation for an arbitrary simplicial set X .

They arise just from a first stage where we add on 1-cells. Namely, if

in doing the procedure Arr(A,m) we replace C by a choice of 1-cell

joining any two components of Am/ which go to the same component

under the Segal map, then applying this operation for various m, we

obtain a simplicial space whose components are connected and homo-

topic to wedges of circles. (We have to start with an X having X1 = ∗).

The resulting simplicial space has the same realization as X . If X has

only finitely many nondegenerate simplices then one can stop after a

finite number of applications of this operation. Taking the fundamental

groups of the component spaces (based at the degeneracy of the unique

basepoint) gives a simplicial free group. Taking the classifying simpli-

cial sets of these groups in each component we obtain a bisimplicial set

whose realization is equivalent to X . This bisimplicial set actually satis-

fies Ap,0 = A0,k = ∗, in other words it satisfies the globular condition in

both directions! We can therefore view it as a Segal precategory in two

ways. The second way, interchanging the two variables, yields a Segal

precategory where the Segal maps are isomorphisms (because at each

stage it was the classifying simplicial set for a group). Thus, viewed in

this way, it is a Segal groupoid and Segal’s theorem implies that the

17.5 Example: π3(S2) 381

simplicial set p 7→ Ap,1, which is the underlying set of a simplicial free

group, has the homotopy type of ΩX .

PART IV

THE MODEL STRUCTURE

18

Sequentially free precategories

In this chapter, we continue the study of weakly enriched categories by

looking at some basic objects: these are the categories with an ordered

set of objects x0, . . . , xn and morphisms other than the identity from xito xj only when i < j. More precisely we consider the free categories

of this type obtained by specifying an object Bi ∈ M of morphisms

from xi−1 to xi for 1 ≤ i ≤ n; then the object of morphisms from xito xj should be the product of the Bk for k = i + 1, . . . , j. One of the

main tasks is to look at a notion of precategory which corresponds to

this notion of category. At the end will be our main calculation, which

is what happens when one takes the product of two such categories.

Throughout, the notion of weak equivalence on PC(X,M ) is the one

given by the model structure of Theorem 14.1.1.

18.1 Imposing the Segal condition on Υ

Recall M -precategories Υ(B1, . . . , Bk) defined in Sections 12.5 and 16.1.

These can be strictly categorified, which is to say that we can construct

corresponding strict M -categories which will be weakly equivalent (The-

orem 18.1.2 below).

Define a precategory Υk(B1, . . . , Bk) as follows: the set of objects

is the same as for Υk(B1, . . . , Bk), that is [k] = υ0, . . . , υk. For any

sequence υi0 , . . . , υin with i0 ≤ . . . ≤ in, we put

Υk(B1, . . . , Bk)(υi0 , . . . , υin) := Bi0+1 ×Bi0+2 × · · · ×Bin−1 ×Bin .

(18.1.1)

This includes the unitality condition Υk(B1, . . . , Bk)(υi, . . . , υi) = ∗. For

any other sequence, that is to say any sequence which is not increasing,


386 Sequentially free precategories

the value is ∅. Note in particular that

Υk(B1, . . . , Bk)(υi−1, υi) = Bi = Υk(B1, . . . , Bk)(υi−1, υi). (18.1.2)

By the adjunction property of Υk(B1, . . . , Bk) there is a unique map

Υk(B1, . . . , Bk) → Υk(B1, . . . , Bk) (18.1.3)

inducing the identity (18.1.2) on adjacent pairs of objects. In fact, we

can consider Υk(B1, . . . , Bk) as a subobject of Υk(B1, . . . , Bk) with this

map as the inclusion.

Lemma 18.1.1 The precategory Υk(B1, . . . , Bk) is a stict M -category,

in particular it is a Segal M -category.

Proof If υi0 , . . . , υin is an increasing sequence i.e. i0 ≤ . . . ≤ in, and if

1 ≤ j ≤ n − 1 then by the formula (18.1.1) the natural maps obtained

by splitting the sequence of objects at υij give an isomorphism

Υk(B1, . . . , Bk)(υi0 , . . . , υin)

Υk(B1, . . . , Bk)(υi0 , . . . , υij )× Υk(B1, . . . , Bk)(υij , . . . , υin).

∼=

↓

By induction it follows that the Segal maps are isomorphisms.

One should think of Υk(B1, . . . , Bk) as being the free M -category

generated by morphism objects Bi going from υi−1 to υi. Here the objects

are linearly ordered, and the generating objects of M are placed between

adjacent objects in the ordering.

We would like to make precise this intuition by showing that it is

the Segal M -category generated by the precategory Υk(B1, . . . , Bk) as

stated in the following theorem.

Theorem 18.1.2 For any k and any sequence of objects B1, . . . , Bk,

the inclusion (18.1.3) is a weak equivalence in PC(υ0, . . . , υk,M ). If

each Bi is cofibrant, then it is a trivial cofibration in the injective model

structure.

18.2 Sequentially free precategories in general

Before getting to the proof of the previous theorem, which will be done

at the end of the chapter on page 395, it is useful to generalize the


above situation by giving a criterion for when a precategory will lead to

a free Segal category with linearly ordered object set and generators Bibetween adjacent objects.

Definition 18.2.1 A sequentially free M -precategory consists of a

finite linearly ordered set X = x0, . . . , xk together with a structure of

M -precategory A ∈ PC(X,M ), satisfying the following properties:

(SF1)—if xi0 , . . . , xin is a sequence of objects which are not increasing,

i.e. there is some 0 < j ≤ n with ij−1 > ij, then A(xi0 , . . . , xin) = ∅;

and

(SF2)—if xi0 , . . . , xin is a sequence of objects in increasing order i.e.

ij−1 ≤ ij for all 0 < j ≤ n, then the outer map for the n-simplex

provides a weak equivalence

A(xi0 , . . . , xin)∼→ A(xi0 , xin). (18.2.1)

Remark 18.2.2 Note that condition (SF2) for a sequence (xi) of

length n = 0 says that the map ∗ = A(xi) → A(xi, xi) is a weak equiv-

alence.

We often say that A is sequentially free with respect to a given order

on X if (X,A) is sequentially free for the ordering in question.

Lemma 18.2.3 Both Υk(B1, . . . , Bk) and Υk(B1, . . . , Bk) are sequen-

tially free M -precategories with respect to the ordering on [k].

Proof Both clearly satisfy (SF1). For Υk(B1, . . . , Bk) the condition

(SF2) holds by construction. For Υk(B1, . . . , Bk), suppose we have an

increasing sequence of objects υi0 ≤ · · · ≤ υin . If in = i0 then the values

on both sides of (18.2.1) are ∗. If in = i0 + 1 then the values on both

sides are Bin , whereas if in > i0 + 1 the values on both sides are ∅. In

all three cases the map (18.2.1) is an isomorphism, a fortiori a weak

equivalence.

Lemma 18.2.4 Suppose X is a fixed linearly ordered finite set, β is

an ordinal, and A(b)b∈β is a transfinite sequence of M -precategories,

that is to say a functor β → PC(X,M ). Suppose that each (X,A(b)) is

a sequentially free M -precategory with respect to the given order on X,

and that for any b ≤ b′ the transition map A(b)→ A(b′) is an injective

cofibration in PC(X,M ). Then the colimit (X, colimb∈βA(b)) is again

a sequentially free M -precategory with respect to the same ordering.

Proof The colimit is calculated levelwise as a ∆oX -diagram in M , since

filtered colimits preserve the unitality condition. A colimit of objects ∅ is


again ∅, so the colimit satisfies (SF1). The left properness hypotheses on

M implies transfinite left properness (Proposition 9.5.3), so the weak

equivalence (18.2.1) is preserved in the colimit whose transition maps

are cofibrations, giving (SF2).

We now come to one of the main steps where we gain some control

over the process of generators and relations. Recall from Section 16.4

the operation Gen consisting of applying one step of the calculus of

generators and relations.

Lemma 18.2.5 Suppose (X,A) is a sequentially free M -precategory

with linearly ordered object set X = x0, . . . , xm, and xa, xa+1, . . . , xbis a strictly increasing sequence of adjacent objects with 0 ≤ a < b ≤ m.

Then the new M -precategory Gen((X,A);xa, . . . , xb) is also sequen-

tially free with the same ordered set of objects X, and furthermore for

any a < j ≤ b the map

A(xj−1, xj) → Gen((X,A);xa, . . . , xb)(xj−1, xj)

is an isomorphism (hence a weak equivalence) in M .

Proof Fix a factorization E, e, p1, . . . , pb−a as used for the construction

of Gen((X,A);xa, . . . , xb). Note that

p : E∼→ A(xa, xa+1)× · · · ×A(xb−1, xb)

is a weak equivalence. The sequence of objects xa, . . . , xb is disjoint, so

we can use the description of Gen((X,A);xa, . . . , xb) given in Lemma

16.4.2. That says that for any sequence of the form xi0 , . . . , xip if a ≤

i0 ≤ · · · ≤ ip ≤ b with i0 + 2 ≤ ip then

Gen(A;xa, . . . , xb)(xi0 , . . . , xip) = A(xi0 , . . . , xip) ∪A(xa,...,xb) E.

(18.2.2)

but for any other sequence,

Gen(A;xa, . . . , xb)(xi0 , . . . , xip) = A(xi0 , . . . , xip). (18.2.3)

We can now check the conditions (SF1) and (SF2). If the sequence of

objects (xi0 , . . . , xip) is not increasing, then it falls into the second case

(18.2.3), and A(xi0 , . . . , xip) = ∅ by (SF1) for A, which gives (SF1) for

Gen(A;xa, . . . , xb). Suppose that (xi0 , . . . , xip) is increasing. We need to

check (SF2). If either i0 < a or ip > b then we again fall into case (18.2.3)

for both the full sequence (xi0 , . . . , xip) and also the pair of endpoints


(xi0 , xip). Thus, by (SF2) for A we have a weak equivalence

Gen(A;xa, . . . , xb)(xi0 , . . . , xip) = A(xi0 , . . . , xip)

A(xi0 , xip) = Gen(A;xa, . . . , xb)(xi0 , xip)

∼

↓

giving this case of (SF2) for Gen(A;xa, . . . , xb). Suppose on the other

hand that a ≤ i0 ≤ ip ≤ b. If ip ≤ i0 + 1 then we again are in case

(18.2.3) for both the full sequence and the pair of endpoints, so we get

the condition (SF2) as above. Suppose therefore that i0 +2 ≤ ip. In the

diagram

A(xi0 , . . . , xip) ← A(xa, . . . , xb) → E

A(xi0 , xip)

↓

← A(xa, . . . , xb)

wwwwwwwww→ E

wwwwwwwwww

the left vertical arrow is an equivalence by (SF2) for A. By left proper-

ness of M via Corollary 9.5.2, the vertical maps therefore induce a weak

equivalence from the pushout of the top row to the pushout of the bot-

tom row. By equations (18.2.2) which apply both to the full sequence

(xi0 , . . . , xip) and the pair of endpoints (xi0 , xip), these pushouts are re-

spectivelyGen(A;xa, . . . , xb)(xi0 , . . . , xip) andGen(A;xa, . . . , xb)(xi0 , xip).

Thus, the map from the one to the other is an equivalence, which gives

condition (SF2) in this last case. This proves thatGen((X,A);xa, . . . , xb)

is again a sequentially free M -precategory.

For the last statement, note that the sequence (xj−1, xj) falls into case

(18.2.3) because the space between the endpoints is only 1. Hence the

formula (18.2.3) says that the map

A(xj−1, xj) → Gen((X,A);xa, . . . , xb)(xj−1, xj)

is an isomorphism.

Recall that the map A → Gen((X,A);xa, . . . , xb) is a weak equiva-

lence in PC(X ;M ).

Lemma 18.2.6 Suppose (X,A) is a sequentially free M -precategory

with ordered object set X = x0, . . . , xm. By iterating a series of oper-

ations of the form A 7→ Gen((X,A);xa, . . . , xb) we can obtain a weak


equivalent sequentially free M -precategory (X,A) → (X,A′) such that

A′ satisfies the Segal conditions.

Proof We show how to obtain the Segal condition for strictly increasing

sequences of adjacent objects. At the end of the proof we go from here

to the Segal condition for general sequences.

Fix an integer n0 and suppose (X,A) satisfies the Segal condition

for all adjacent sequences (xc, xc+1, . . . , xd) with d − c > n0, and a

certain number of adjacent sequences (xcv , xcv+1, . . . , xdv ) for (cv, dv)

indexed by v ∈ V for some set v, with dv − cv = n0. Suppose 0 ≤

a < b ≤ m with b − a = n0. Then Gen((X,A);xa, . . . , xb) satisfies the

Segal condition for all adjacent sequences (xc, xc+1, . . . , xd) with d− c >

n0, for the given adjacent sequences (xcv , xcv+1, . . . , xdv ) with v ∈ V ,

and also for the sequence (xa, . . . , xb). Indeed, if (xc, xc+1, . . . , xd) with

d− c > n0 then the terms entering into the Segal map for this sequence

are all covered by the situation (18.2.3) in the explicit description of

Gen((X,A);xa, . . . , xb) used in the previous proof (see Lemma 16.4.2).

Note that the terms in the product on the right hand side of the Segal

map are of the form Gen((X,A);xa, . . . , xb)(xj−1, xj) = A(xj−1, xj).

Here we use the condition that we are only looking at adjacent sequences.

By the recurrence hypothesis on A, the Segal map for this sequence is

an equivalence.

Similarly, if (xcv , xcv+1, . . . , xdv ) is one of our given sequences with

dv − cv = n0, and if it is different from the sequence (xa, . . . , xb),

then either cv < a or dv > b so again everything entering into the

Segal map for this sequence is the same as for A, by (18.2.3). Thus,

again the inductive hypothesis on A implies the Segal condition for

Gen((X,A);xa, . . . , xb)(xcv , xcv+1, . . . , xdv ).

At the sequence (xa, . . . , xb) equation (18.2.2) gives

Gen(A;xa, . . . , xb)(xa, . . . , xb) = A(xa, . . . , xb) ∪A(xa,...,xb) E = E

and the map

E → A(xa, xa+1)× · · · ×A(xb−1, xb)

is an equivalence by hypothesis on the choice of E. Therefore, the Segal

condition holds at the sequence (xa, . . . , xb) too and we can add this to

our collection V of good sequences of lenght n0.

So, the inductive procedure is to start with the maximal sequence

x0, . . . , xm of lengthm, impose the Segal condition here by changingA to

Gen(A;x0, . . . , xm); then successively impose the Segal condition on all


sequences of lengthm−1, thenm−2 and so on, using the above inductive

observation. At each step A is replaced by Gen(A;xa, . . . , xb) and there

is a map from the old to the new A which is a weak equivalence in

PC(X,M ). By Lemma 18.2.5, the new A is always again a sequentially

free M -precategory. Combining these steps (there are a finite number)

down to n0 = 2 we arrive at a map (X,A) → (X,A′) which is a weak

equivalence in PC(X,M ), and such that A′ is a sequentially free M -

precategory which satisfies the Segal condition for all sequences of the

form (xc, . . . , xd).

We claim that this implies that A′ satisfies the Segal condition for any

sequence of the form xi0 , . . . , xip . Note that if p = 0 then the Segal con-

dition is automatic since A′(X) = ∗. Suppose p ≥ 1 and consider our se-

quence (xi0 , . . . , xip−1 , xip). If any ij−1 > ij then A′(xi0 , . . . , xip−1 , xip) =

∅ and A′(xij−1 , xij ) = ∅. The second statement, plus Lemma 10.0.10 on

the direct product with ∅, imply that the right hand side of the Segal

map is ∅, which is the same as the left side by the first statement. So in

this case, the Segal condition is automatic. Hence we may assume that

ij−1 ≤ ij for all 1 ≤ j ≤ p.

Let a := i0 and b := ip, and denote by (xa, . . . , xb) the full sequence

of adjacent elements (each counted once) going from xa to xb. There

is a unique map σ : (xi0 , . . . , xip−1 , xip) → (xa, . . . , xb) in ∆X , sending

each xij to the same object at the unique place it occurs in (xa, . . . , xb).

Note that we are using here the reduction of the previous paragraph

that ij−1 ≤ ij for all 1 ≤ j ≤ p, guaranteeing that each xij occurs in the

list (xa, . . . , xb). This σ induces a map

σ∗ : A′(xa, . . . , xb)∼→ A′(xi0 , . . . , xip−1 , xip). (18.2.4)

That this is a weak equivalence, can be seen by considering the diagram

with maps to the spanning pair (xa, xb) = (xi0 , xip):

A′(xa, . . . , xb) → A′(xi0 , . . . , xip−1 , xip)

A′(xa, xb)

↓

======== A′(xi0 , xip).

↓

The vertical maps are weak equivalences by the sequentially free condi-

tion, so the top map is a weak equivalence by 3 for 2.

On the other hand, for each pair in the original sequence, consider the


Segal map for it:

σi0,i1 : A′(xi0 , . . . , xi1 ) → A′(xi0 , xi0+1)× · · · ×A′(xi1−1, xi1),

...

σip−1,ip : A′(xip−1 , . . . , xip) → A′(xip−1 , xip−1+1)× · · · ×A′(xip−1, xip).

The mini-sequences appearing on the left hand sides, are the sequences

of length ij− ij−1 going from ij−1 to ij by intervals of step 1. Whenever

ij−1 = ij, we have a sequence of length zero and both sides of the map

are equal to ∗. The case of the Segal condition which we already know,

says that the σij−1,ij are weak equivalences.

Putting these all together, we get a weak equivalence whose target is

the full Segal product for the sequence (xa, . . . , xb) considered above:

A′(xi0 , . . . , xi1 )×· · ·×A′(xip−1 , . . . , xip)

∼→ A′(xa, xa+1)×· · ·×A

′(xb−1, xb).

On the other hand, the map to the spanning interval gives a map for

each mini-sequence

A′(xij−1 , . . . , xij )∼→ A′(xij−1 , xij ) (18.2.5)

which is a weak equivalence, by the condition that A′ is sequentially free.

The mini-sequences map into the full adjacent sequence (xa, . . . , xb),

giving maps

A′(xa, . . . , xb) → A′(xij−1 , . . . , xij ).

The Segal map for (xa, . . . , xb) thus factors as

A′(xa, . . . , xb)∼→ A′(xi0 , . . . , xi1)× · · · ×A

′(xip−1 , . . . , xip)

A′(xa, xa+1)× · · · ×A′(xb−1, xb).

∼

↓

(18.2.6)

The second arrow is a weak equivalence as pointed out previously, and

the composition is a weak equivalence by the Segal condition for (xa, . . . , xb)

which we already know, so the first map is a weak equivalence by 3 for

2.


Now we have a commutative diagram

A′(xa, . . . , xb) → A′(xi0 , . . . , xip−1 , xip)

A′(xi0 , . . . , xi1 )× · · · ×A′(xip−1 , . . . , xip)

↓

→ A′(xi0 , xi1 )× · · · ×A′(xip−1 , xip)

↓

where the top arrow is a weak equivalence as seen above (18.2.4), the

left vertical map is a weak equivalences by (18.2.6), and the bottom map

is a weak equivalence by combining together the equivalences (18.2.5).

By 3 for 2, it follows that the right vertical map is a weak equivalence,

which is the Segal condition for the sequence (xi0 , . . . , xip−1 , xip). This

completes the proof of the lemma.

Corollary 18.2.7 Suppose (X,A) is a sequentially free ordered M -

precategory, and let r : (X,A) → (X,A′) be a fibrant replacement in

either the projective or injective model structure on PC(X ;M ) con-

structed in Theorem 14.1.1. Then (X,A′) is a sequentially free ordered

M -precategory for the same order on X, and for any 0 < j ≤ n the map

A(xj−1, xj) → A′(xj−1, xj) is a weak equivalence.

The same conclusions holds if, instead of a fibrant replacement, r is a

weak equivalence to an object A′ which satisfies the Segal conditions.

Proof Use either the projective or the injective structure in what fol-

lows. Note that the sequentially free condition is preserved by levelwise

weak equivalences of unital diagrams on ∆oX .

By Lemma 18.2.6, there is a map A → A′′, weak equivalence in

PC(X,M ), such that A′′ satisfies the Segal conditions and is still se-

quentially free. Consider furthermore a map s : A′′ → A3 which is a

trivial cofibration to a fibrant object in PC(X,M ). Then A3 also sat-

isfies the Segal conditions, so Lemma 14.4.1 says that s is a levelwise

weak equivalence. It follows that A3 is again sequentially free.

Our different fibrant replacement r : A→ A′ is a trivial cofibration, so

there is a map g : A′ → A3 compatible with the maps from A. By 3 for

2 this map g is a weak equivalence in PC(X,M ), so again by Lemma

14.4.1 it is a levelwise weak equivalence, hence A′ is sequentially free.

Furthermore, A(xj−1, xj) → A′′(xj−1, xj) is a weak equivalence in M ;

it follows from the above levelwise weak equivalences compatible with

the maps from A, that the same is true for A3, then A′.

For the last paragraph, suppose r : A → A′ is a weak equivalence in


PC(X,M ) and A′ satisfies the Segal conditions. Then choosing first a

fibrant replacement A3 of A′, then a factorization, we can get a square

A → A′

A′′

↓

→ A3

↓

such that the vertical arrows are fibrant replacements and all arrows

are weak equivalences. By Lemma 14.4.1, the right vertical and bottom

maps are levelwise weak equivalences since A′ and A′′ satisfy the Segal

conditions. By the first part of the present lemma, A′′ is sequentially

free, it follows that A3 and then A′ are sequentially free. On adjacent

pairs of objects the left vertical map induces an equivalence, so by 3 for

2 the top map does too. This shows that the required statements hold

for A′.

In the situation of the previous corollary, suppose (xi0 , . . . , xin) is an

increasing sequence of objects. Then we have weak equivalences

A′(xi0 , xin)∼→ A′(xi0 , . . . , xin)

∼→ A′(xi0 , xi1 )× · · · ×A

′(xin−1 , xin).

On the other hand if the sequence is not increasing, the corresponding

morphism space is ∅.

Suppose that our sequence of objects is a sequence of adjacent objects,

that is to say look at a sequence of the form (xa, xa+1, . . . , xb) for a ≤ b.

The condition of the corollary says furthermore that we have weak equiv-

alences A(xj−1, xj)∼→ A′(xj−1, xj) for a < j ≤ b. These equivalences

go together to give an equivalence on the level of the direct product, by

the product conditions (PROD) and (DCL) for M as in Lemma 10.0.11.

Thus, the pair of weak equivalences in the previous paragraph extends

to a chain of equivalences

A′(xa, . . . , xb)∼→ A′(xa, xa+1)× · · · ×A

′(xb−1, xb)

A′(xa, xb)

∼

↓

A(xa, xa+1)× · · · ×A(xb−1, xb).

∼

↑

(18.2.7)

All in all, this says that (X,A′) looks up to homotopy very much like

the Υk(B1, . . . , Bk) defined at the start, with Bi = A(xi−1, xi).


Corollary 18.2.8 Suppose f : (X,A) → (X,B) is a morphism in

PC(X ;M ) between M -precategories which are both sequentially free for

the same ordering of the underlying set of objects X. Suppose that for any

adjacent objects xj−1, xj in the ordering, f induces a weak equivalence

A(xj−1, xj)∼→ B(xj−1, xj). Then f is a weak equivalence in PC(X ;M ).

Proof View the object set as numbered X = x0, . . . , xk. Let (X,A′)

and (X,B′) be fibrant replacements for (X,A) and (X,B) respectively,

and we may assume that f extends to a map f ′ : (X,A′) → (X,B′).

For any 0 ≤ a ≤ b ≤ k, f ′ and f induce a morphism between the

chains of equivalences considered in (18.2.7), for A,A′ and B,B′. The

hypothesis of the present corollary says that the induced map on the

bottom right corner is a weak equivalence; thus the induced map f ′ :

A′(xa, xb) → B′(xa, xb) is a weak equivalence. Note on the other hand

that if a > b then A′(xa, xb) = ∅ and B′(xa, xb) = ∅. Thus f

′ induces a

weak equivalence on the morphism space for any pair of objects. By the

Segal condition, it follows that f ′ is an objectwise weak equivalence in

the category of ∆oX -diagrams in M , so it is a weak equivalence in the

model structures. Since f ′ is a fibrant replacement for f , this implies

that f was a weak equivalence.

Proof of Theorem 18.1.2: the map

f : Υk(B1, . . . , Bk) → Υk(B1, . . . , Bk)

satisfies the hyptheses of the previous corollary, so we conclude that it is

a weak equivalence. If the Bi are cofibrant, then the map in question is a

cofibration in the injective model structure, because is is an objectwise

cofibration when viewed in the injective model structure: the induced

maps are either identities, or maps from ∅ into a product of copies of Bi.

A product of copies of Bi is cofibrant, because of the cartesian conditions

on M (see Lemma 10.0.12).

19

Products

In this chapter, we consider the direct product of two M -enriched pre-

categories. On the one hand, we would like to maintain the cartesian

or product condition (PROD) for the new model category we are con-

structing. On the other hand, compatibility with direct product pro-

vides the main technical tool we need in order to study pushouts by

weak equivalences in PC(M ). Recall that we already have the injective

and projective model structures of Theorem 14.1.1 on PC(X,M ) at our

disposal.

Here is the place where we really use the full structure of the category

∆, as well as the unitality condition. At the end of the chapter, we’ll

discuss some counterexamples showing why these aspects are necessary.

19.1 Products of sequentially free precategories

Suppose X = x0, . . . , xm and Y = y0, . . . , yn are finite linearly or-

dered sets, and (X,A) and (Y,B) are sequentially free M -precategories,

with respect to the orderings. The product of these objects considered

in the category PC(M ), has the form (X × Y,A⊠ B) where

(A⊠ B)((xi0 , yj0), . . . , (xip , yjp) := A(xi0 , . . . , xip)× B(yj0 , . . . , yjp).

Let A → A′ and B → B′ be weak equivalences towards objects satis-

fying the Segal condition and which are therefore also sequentially free

(Corollary 18.2.7). We obtain a map of M -precategoriesA⊠B → A′⊠B′

on the object set X×Y . We would like to show that this is a weak equiv-

alence in PC(X × Y ;M ).

For any subset S ⊂ X×Y , let i(S) : S → X×Y denote the inclusion



map, and

i(S)∗ : PC(X × Y,M ) → PC(S,M )

the pullback map on M -precategory structures. This can be pushed back

to a M -precategory structure with object set X × Y , and the resulting

functor will be denoted as i(S)∗! for brevity:

i(S)∗! := i(S)!i(S)∗ : PC(X × Y,M ) → PC(X × Y,M ).

This may be described explicitly as follows. Suppose C ∈ PC(X ×

Y,M ). Recall from the first paragraph of Section 12.3, that i(S)!i(S)∗(C)

is the same as the precategory i(S)∗(C) over set of objects S, extended

by adding on the discrete M -enriched category on the complementary

set X×Y −S. Thus, for any sequence of objects ((xi0 , yi0), . . . , (xip , yip))

we have:

i(S)∗! (C)((xi0 , yi0), . . . , (xip , yip)) = C((xi0 , yi0), . . . , (xip , yip))

if all the (xij , yij ) are in S; if any of the pairs is not in S and the sequence

is not constant then

i(S)∗! (C)((xi0 , yi0), . . . , (xip , yip)) = ∅,

while the unitality condition requires that

i(S)∗! (C)((xi0 , yi0), . . . , (xi0 , yi0)) = ∗,

for a constant sequence at any point (xi0 , yi0) ∈ X × Y .

Lemma 19.1.1 If C satisfies the Segal condition, then so does i(S)∗(C)

(on object set S) and i(S)∗! (C) (on object set X × Y ).

Proof This follows immediately from Lemma 12.3.1.

The above discussion is usually applied to an object of the form C =

A⊠ B ∈ PC(X × Y ;M ).

Here are the special types of subsets S we will consider. A box will be

a subset of the form

Ba,b := x0, . . . , xa × y0, . . . , yb ⊂ X × Y,

398 Products

which can be pictured for example with (a, b) = (5, 3) as

r r r r r r

r r r r r r

r r r r r r

r r r r r r

(x0, y0)

(x0, yb)

(xa, y0)

(xa, yb)

A notched box is a subset

Ba,b ⊃ Bνa,b := Ba,b−1 ∪ (xν , yb), . . . , (xa, yb),

defined when b ≥ 1. Thus, a notched box Bνa,b is a box Ba,b minus a

part of the top row (x0, yb), . . . , (xν−1, yb). For example a notched box

with (a, b) = (5, 3) and ν = 3 looks like

r r r r r r

r r r r r r

r r r r r r

r r r

(x0, y0)

(x0, yb−1)

(xa, y0)

(xν , yb)

(xν , yb−1)

(xa, yb)

The boxes and notched boxes are defined for 0 ≤ a ≤ m and 0 ≤ b ≤ n,

and 0 ≤ ν ≤ a.

If a = m, b = n then Ba,b = X × Y , and if ν = 0 then Bνa,b = Ba,b.

For any ν, the upper right corner of Bνa,b is equal to the point (xa, yb).

The lower left corner is (x0, y0). The corner of the notch of Bνa,b is the

point (xν , yb). Note that

Bνa,b − (xν , yb) = Bν+1

a,b


whenever 0 ≤ ν < a. For ν = a we have Bνa,b − (xν , yb) = Ba,b−1.

A tail is a subset of the form

Ti0,...,ipj0,...,jp

:= (xi0 , yj0), . . . , (xip , yjp)

which will be considered whenever

i0 ≤ i1 ≤ · · · ≤ ip, iu ≤ iu−1 + 1,

j0 ≤ j1 ≤ · · · ≤ jp, ju ≤ ju−1 + 1,

and (iu, ju) 6= (iu−1, ju−1). Put another way, at each place in the pair of

sequences, there are three possibilities for (iu, ju) in terms of (iu−1, ju−1):

(iu, ju) = (iu−1 + 1, ju−1) or (iu−1, ju−1 + 1) or (iu−1 + 1, ju−1 + 1).

Thus, the sequence moves in single horizontal, vertical or diagonal steps.

We say that the tail goes from (xi0 , yj0) to (xip , yjp). A full tail is one

which goes from (x0, y0) to (xm, yn).

A dipper is a subset S which is the union of a notched box Bνa,b, with

a tail Ti0,...,ipj0,...,jp

going from (xi0 , yj0) = (xa, yb) to (xip , yjp) = (xm, yn).

These overlap at the point (xa, yb). For example, a dipper starting with

the previous notched box, and going out to (m,n) = (9, 7) could look

like this:

q q q q q q

q q q q q q

q q q q q q

q q q

(xa, yb)

^(xν , yb)

~q

qq q

qq

(xm, yn)

For b ≥ 1, the union of Bba,b with a tail T going from (xa, yb) to

(xm, yn), is equal to the union of B0a,b−1 with a tail T ′ = (xa, xb−1)

going from (xa, xb−1) to (xm, yn), with the union overlapping at the

point (xa, xb−1). For example, in the previous picture if we set ν = b it

400 Products

becomes

q q q q q q

q q q q q q

q q q q q q

q

(xa, yb)

^

(xa, yb−1)q

qq q

qq

(xm, yn)

In this situation if furthermore b−1 = 0, then the subset S becomes a

full tail going from (x0, y0) to (xm, yn). Note also that a box with a = 0

plus a tail, is again equal to a full tail.

In view of this, we consider the variables a, b, ν for the notched box in

a dipper S, only in the range 0 ≤ ν < a and 0 < b. The corner of the

notch (xν , yb) is well-defined, and S−(xν , yb) is again either a dipper,

or a full tail.

The product of the linear orders onX and Y is a partial order onX×Y

where (xa, yb) < (xa′ , yb′) whenever a ≤ a′, b ≤ b′ and (a, b) 6= (a′, b′).

Given a dipper S, let (xν , yb) be the corner of the notch and define new

subsets

S>(xν ,yb)< := (x′, y′) ∈ S, either (x′, y′) < (xν , yb) or (xν , yb) < (x′, y′)

and

S≥(xν ,yb)≤ := (x′, y′) ∈ S, either (x′, y′) ≤ (xν , yb) or (xν , yb) ≥ (x′, y′).

Note that

S≥(xν ,yb)≤ = S>(xν ,yb)< ∪ (xν , yb).


For example, for the dipper pictured previously with a = 5, b = 3, ν =

3,m = 9, n = 7, we have the picture for S>(xν ,yb)<

q q q q

q q q q

q q q q

qq

(xa, yb)

^(xν+1, yb)

q

(xν , yb−1)q

qq q

qq

(xm, yn)

and the picture for S≥(xν ,yb)≤

q q q q

q q q q

q q q q

qqq

(xa, yb)

^(xν , yb)

~q

qq q

qq

(xm, yn)

Lemma 19.1.2 If S is a dipper with notched box Bνa,b for 0 ≤ ν <

a and 0 < b, then either ν > 0 and b > 1 in which case S>(xν ,yb)<

and S≥(xν ,yb)≤ are dippers with notched box B0ν,b−1 and tails going from

(xν , yb−1) to (xm, yn); or else either ν = 0 or b = 1 in which case

S>(xν ,yb)< and S≥(xν ,yb)≤ are full tails.

Proof Look at the above pictures.

The idea of the proof of the product property is to consider subsets S

which are dippers or full tails, and prove that i(S)∗(A⊠B) → i(S)∗(A′⊠

B′) is a weak equivalence in PC(S;M ).

We first discuss the case of a full tail. Then the case of dippers

will be treated by induction on a, b, ν, eventually getting to the case

a = m, b = n which gives the desired theorem. The case of full tails,

treated in the following proposition, encloses the case of all increasing

single-step paths going from (0, 0) to (m,n). This formalizes the intuition

402 Products

that to understand the product we should understand what happens on

each path. This is similar to what is going on in the decomposition of

a product of simplices: the product has a decomposition into simplices

indexed by the same collection of paths—although the simplices of max-

imal dimension correspond to paths without diagonal steps.

Proposition 19.1.3 Suppose (X,A) and (Y,B) are sequentially free

M -precategories. Suppose T = Ti0,...,ipj0,...,jp

is a full tail. The induced or-

dering on T is a linear order, and i(T )∗(A ⊠ B) is a sequentially free

M -precategory on the linearly ordered object set T . Furthermore, sup-

pose A → A′ and B → B′ are weak equivalences towards sequentially

free precategories satisfying the Segal condition, in the model categories

of Theorem 14.1.1. Then

i(T )∗(A⊠ B) → i(T )∗(A′⊠ B′)

is a weak equivalence whose target satisfies the Segal condition. The same

is also true of i(T )∗! (A⊠ B) whose object set is X × Y .

Proof An increasing sequence of objects of T has the form (z0, . . . , zr)

where zk = (xu(k), yv(k)) with u(k) = ia(k) and v(k) = ja(k) for 0 ≤

a(0) ≤ . . . ≤ a(r) ≤ p an increasing sequence in the set of indices for the

objects of T . The tail condition means that for any such increasing se-

quence, the sequences xu(0), . . . , xu(r) and yu(0), . . . , yu(r) are increasing

sequences in X and Y respectively. Now

i(T )∗(A⊠ B)(z0, . . . , zr) = A(xu(0), . . . , xu(r))× B(yu(0), . . . , yu(r))

whereas

i(T )∗(A⊠ B)(z0, zr) = A(xu(0), xu(r))× B(yu(0), yu(r))

so the fact that A and B are sequentially free implies, via the cartesian

condition for M , that the map

i(T )∗(A⊠ B)(z0, . . . , zr) → i(T )∗(A⊠ B)(z0, zr)


Suppose A → A′ and B → B′ are weak equivalences towards sequen-

tially free precategories satisfying the Segal condition. It follows from

Corollary 18.2.7 that for any adjacent pair of objects xi−1, xi ∈ X , the

map

A(xi−1, xi) → A′(xi−1, xi)

is a weak equivalence. The sequentially free condition implies thatA(xi, xi)


and A′(xi, xi) are contractible (see Remark 18.2.2). In particular for any

object xi the map

A(xi, xi) → A′(xi, xi)

is a weak equivalence. The same two statements hold for B. But now

the tail property of T says that an adjacent pair of objects in T is of the

form (xi, yj), (xk, yl) where xi, xk are either adjacent objects or the same

object in X , and yj , yl are either adjacent objects or the same object in

Y . It follows that the map

A(xi, xi)× B(yj, yl) → A′(xi, xi)× B

′(yj , yl)

is a weak equivalence. By Corollary 18.2.8, the map

i(T )∗(A⊠ B) → i(T )∗(A′⊠ B′)

is a weak equivalence. The set of objects of T injects into X × Y , so

upon pushing forward to precategories on object set X × Y , again

i(T )∗! (A⊠ B) → i(T )∗! (A′⊠ B′)


The next step is to note that when we remove the corner of the

notch from a dipper S we have a pushout diagram. In order to give

the statement with some generality, say that C ∈ PC(X × Y,M ) is or-

dered if C((xi0 , yj0), . . . , (xip , yjp)) = ∅ whenever we have a nonincreas-

ing sequence (xi0 , yj0), . . . , (xip , yjp) in the product order, that is to say

whenever there is some k such that either ik−1 > ik or jk−1 > jk. Let

PCord(X × Y ;M ) denote the category of ordered M -enriched precate-

gories over X × Y .

Proposition 19.1.4 Suppose S = Bνa,b ∪ T is a dipper and (xν , yb)

the corner of the notch. Assume 0 ≤ ν < a and 0 < b. Then the pushout

expression for subsets of X × Y

S = (S − (xν , yb)) ∪S>(xν,yb)< S≥(xν ,yb)≤

extends to a pushout expression for any ordered C ∈ PCord(X × Y ;M ):

the square

i(S>(xν ,yb)<)∗! (C) → i(S≥(xν ,yb)≤)

∗! (C)

i(S − (xν , yb))∗! (C)

↓

→ i(S)∗! (C)

↓

404 Products

is a pushout square in the category PC(X × Y ;M ).

Proof It suffices to verify this levelwise on ∆X×Y , in other words we

have to verify it for every sequence of objects (xi0 , yi0), . . . , (xip , yip). If

either of x· or y· is not increasing then it is trivially true, so we may

assume that both are increasing. If none of the elements in the sequence

are (xν , yb) then it is again trivially true. So we may assume that there

is a j with (xij , yij ) = (xν , yb). Then the full sequence is contained in

the region S≥(xν ,yb)≤. It follows that both vertical maps in the above

diagram, at the level of the sequence (x·, y·), are isomorphisms. This

implies that the diagram is a pushout.

Looking at this before putting everything back onto the same object

set X × Y , one can also say that the square

i(S>(xν ,yb)<)∗(C) → i(S≥(xν ,yb)≤)

∗(C)

i(S − (xν , yb))∗(C)

↓

→ i(S)∗(C)

↓

is a pushout square in PC(M ). This version of the statement perhaps

explains better what is going on, but is less useful to us since we are

currently working in the model category structure on PC(X × Y ;M )

for a fixed set of objects X × Y .

Corollary 19.1.5 Suppose g : C → C′ is a map of ordered M -enriched

precategories over object set X×Y , such that both C and C′ are levelwise

cofibrant, that is cofibrant in PCinj(X×Y,M ). Suppose that for any full

tail T going from (0, 0) to (m,n) the map

i(T )∗! (C) → i(T )∗! (C′)

is a weak equivalence. Then g is a weak equivalence.

Proof We show by induction that for any dipper of the form S = Bνa,b∪

T , the map

i(S)∗! (C) → i(S)∗! (C′)

is a weak equivalence. The induction is by (b, a − ν) in lexicographic

order. In the initial case b = 1 and ν = a, S is a tail so the inductive

statement is one of the hypotheses. Suppose the statement is known for

all dippers S′ corresponding to (a′, b′, ν′) with b′ < b or b′ = b and

a′ − ν′ < a − ν. Use the expression of Proposition 19.1.4: i(S)∗! (C) and


i(S)∗! (C′) are respectively the pushouts of the top and bottom rows in

the diagram

i(S − (xν , yb))∗! (C) ← i(S>(xν ,yb)<)

∗! (C) → i(S≥(xν ,yb)≤)

∗! (C)

i(S − (xν , yb))∗! (C

′)

↓

← i(S>(xν ,yb)<)∗! (C

′)

↓

→ i(S≥(xν ,yb)≤)∗! (C

′).

↓

The vertical maps in the diagram induce the given map i(S)∗! (C) → i(S)∗! (C′).

The horizontal maps are cofibrations in the injective model structure

PCinj(X × Y,M ). Indeed, on any sequence (xi0 , yj0), . . . , (xip , yjp) of

points in X×Y , each of the horizontal maps is either the identity, or the

inclusion from ∅ to C(xi0 , yj0), . . . , (xip , yjp) or C′(xi0 , yj0), . . . , (xip , yjp).

By hypothesis, C and C′ are levelwise cofibrant, so the inclusions from ∅

are cofibrations. This shows that the horizontal maps are cofibrations.

The inductive hypothesis applies to each of the vertical maps. This

is seen by noting that the invariants (b′, a′ − ν′) for the dippers S −

(xν , yb), S>(xν ,yb)< and S≥(xν ,yb)≤ are strictly smaller than (b, a− ν)

in lexicographic order:

—for S−(xν , yb) we have a′ = a, ν′ = ν+1 and b′ = b, unless ν = a−1

in which case b′ = b− 1;

—for S>(xν ,yb)< and S≥(xν ,yb)≤ we have b′ = b− 1.

Hence, by the inductive hypothesis, each of the vertical maps is a

weak equivalence. Now PCinj(X × Y,M ) is left proper because it is a

left Bousfield localization cf Theorem 14.1.1, which implies by Corol-

lary 9.5.2 that cofibrant pushouts are preserved by weak equivalences

(see alternatively Lemma 16.3.4). This shows that the map on pushouts

i(S)∗! (C) → i(S)∗! (C′) is a weak equivalence. This completes the inductive

proof.

At the last case S = X × Y we obtain the conclusion of the corollary,

that g : C → C′ is a weak equivalence.

This gives the first main result of this chapter.

Theorem 19.1.6 Suppose (X,A) and (Y,B) are sequentially free M -

precategories, cofibrant in PCinj(X × Y,M ). Suppose A → A′ and

B → B′ are trivial cofibrations in PCinj(X,M ) and PCinj(Y,M ) re-

spectively, towards sequentially free M -precategories. Then the map

(X × Y,A⊠ B) → (X × Y,A′⊠ B′)

is a trivial cofibration in PCinj(X × Y,M ).

406 Products

Proof Suppose first that A′ and B′ satisfy the Segal condition. In this

case, Corollary 19.1.5 applies with C = A⊠B and C′ = A′⊠B′. The hy-

pothesis on full tails used in Corollary 19.1.5 is provided by Proposition

19.1.3.

If A′ and B′ do not themselves satisfy the Segal condition, we can

choose further morphisms A′ → A′′ and B′ → B′′ which are trivial

cofibrations in PCinj(X,M ) and PCinj(Y,M ) respectively, such that

A′′ and B′′ satisfy the Segal condition. The first case of this proof then

applies to the maps from A and B, and also to the maps from A′ and

B′. These show that

(X × Y,A⊠ B) → (X × Y,A′′⊠ B′′)

and

(X × Y,A′⊠ B′) → (X × Y,A′′

⊠ B′′)

are trivial cofibrations in PCinj(X × Y,M ). By 3 for 2 it follows that

(X × Y,A⊠ B) → (X × Y,A′⊠ B′)

is a weak equivalence in PCinj(X × Y,M ), and it is a trivial cofibration

by the cartesian property of M (conditions (PROD) and (DCL), applied

as in Lemma 10.0.12).

19.2 Products of general precategories

The next step is to extend the result of Theorem 19.1.6 from the sequen-

tially free case, to the product of arbitrary M -enriched precategories.

Recall that we have defined morphisms

Υk(B1, . . . , Bk) → Υk(B1, . . . , Bk)

of sequentially free M -precategories, the target satisfies the Segal con-

ditions, and the morphism is a weak equivalence in PC([k],M ) by The-

orem 18.1.2. Use the notation

Σ([k];B) := Υk(B, . . . , B)

where the same object B occurs k times.

Lemma 19.2.1 In the case where B1 = . . . = Bk = B ∈M , the map

Υ → Υ factors as

Σ([k];B) → h([k];B) → Υk(B, . . . , B),


and both maps are global weak equivalences between sequentially free M -

enriched precategories.

Proof All three are sequentially free, and Corollary 18.2.8 applies. For

Σ([k];B) and Υk(B, . . . , B) this is exactly what we said in the proof

of Theorem 18.1.2; for h([k];B) see the explicit description in Section

12.5.

Proposition 19.2.2 Suppose A ∈ PC(M ). Then we can obtain a

global weak equivalence A → A′ such that A′ satisfies the Segal condi-

tions i.e. A′ ∈ R, by taking a transfinite composition of pushouts along

morphisms of the form

Σ([k];V ) ∪Σ([k];U) h([k];U) → h([k];V ) (19.2.1)

for generating cofibrations Uf→ V in M .

Proof The maps (19.2.1) are the same as the Ψ([k], f) considered in

Corollary 16.2.5 and which make up the new pieces in KReedy. Note

that this collection is missing the piece of KReedy consisting of the gen-

erators for levelwise trivial Reedy cofibrations. However, for the present

statement that piece is not needed: if we apply the small object argument

to the present collection of morphisms we can obtain a map A → A′

which is a transfinite composition of pushouts along morphisms of the

form (19.2.1), such that A′ satisfies the left lifting property with respect

to this collection. The pushouts in question preserve weak equivalences,

indeed the maps are a part of KReedy so that follows from the construc-

tion of the model structure of Theorem 14.3.2 by direct left Bousfield

localization; or else one could apply Theorem 16.3.3 and Lemma 16.3.5

which is really saying pretty much the same thing. Now, an object which

satisfies the left lifting property with respect to the Ψ([k], f), satisfies

the Segal conditions because the product maps satisfy lifting along any

generating cofibration U → V for M , thus they are trivial fibrations in

M , which shows that A′ satisfies the Segal condition.

Notice that in the construction of the proposition, the resulting A′

will not in general be even levelwise fibrant, one would have to include

pushouts along morphisms of the form h([k], f) for f a generating trivial

cofibration of M .

Theorem 19.2.3 Suppose A ∈ PC(M ) is Reedy cofibrant, k ∈ N and

B ∈M is a cofibrant object. Then the map

A× Σ([k];B) → A× h([k];B)

408 Products


Proof The proof goes in several steps.

(i) Suppose A is a sequentially free M -enriched precategory. Note that

Σ([k];B) → h([k];B) is a trivial cofibration of sequentially free M -

enriched precategories, inducing an order-preserving isomorphism on

objects. Apply Theorem 19.1.6 to this map and the identity of A, to

conclude that

A× Σ([k];B) → A× h([k];B)

is a global weak equivalence. This completes the proof when A is se-

quentially free. This applies in particular to the h([k];B) which are se-

quentially free.

(ii) Recall from Theorem 16.3.3, that we already know that any pushout

along a global trivial cofibration inducing an isomorphism on sets of

objects, is again a global trivial cofibration.

(iii) Suppose we know the statement of the theorem for A, A′ and A′′

and suppose given a diagram in which one of the arrows is at least an

injective (i.e. levelwise) cofibration

A′ ← A → A′′,

then we claim that the statement of the theorem is true for Q := A′ ∪A

A′′. Indeed,

Q× Σ([k];B) = (A′ × Σ([k];B)) ∪A×Σ([k];B) (A′′ × Σ([k];B))

by commutation of colimits and direct products in PC(M ) (which is

part (DCL) of the cartesian condition 10.0.9). The same is true for the

product with h([k];B). The map

Q× Σ([k];B) → Q× h([k];B) (19.2.2)


is therefore obtained by functoriality of the pushout of the columns in

A′ × Σ([k];B) → A′ × h([k];B)

A× Σ([k];B)

↑

→ A× h([k];B)

↑

A′′ × Σ([k];B)

↓

→ A′′ × h([k];B).

↓

We are supposing that we know that each of the horizontal maps is

a global weak equivalence, also they induce isomorphisms on objects.

By the cartesian property for M applied levelwise, the same one of the

vertical maps is a levelwise cofibration.

By Lemma 16.3.4, the induced map on pushouts (19.2.2) is a global

weak equivalence. This proves the claim for step (iii).

(iv) Suppose given a sequenceAi indexed by an ordinal β, with injective

cofibrant transition maps. Suppose the statement of the theorem is true

for each Ai, then it is true for Q := colimi∈βAi. Indeed, just as in the

previous part Q×h([k];B) can be expressed as a transfinite composition

of pushouts of Q×Σ([k];B) along maps which are by hypothesis global

trivial cofibrations which induce isomorphisms on objects. By Theorem

16.3.3 and Lemma 16.3.5, the composition is a global weak equivalence.

(v) We show by induction on m ∈ N that if skm(A) ∼= A then the

statement of the theorem holds for A. It is easy to see in case m = 0

because then A is just a discrete set. Suppose this is known for any

m ≤ n, and suppose A = skn(A). By Proposition 15.5.1 we can express

A as a transfinite composition of pushouts of skn−1(A) along maps of

the form h([n], ∂[n];U → V ) → h([n];V ). On the other hand,

h([n], ∂[n];U → V ) = h([n];U) ∪h(∂[n];U) ∂h(∂[n];V ),

and h(∂[n];U) = skn−1h([n];U). By the inductive hypothesis the state-

ment of the theorem is known for h(∂[n];U) and similarly for h(∂[n];V ).

It is known for h([n];U) by (i). So by (iii) the statement of the theorem

is known for h([n], ∂[n];U → V ). Furthermore it is known for skn−1(A)

by the inductive hypothesis. Again by (iii) and (iv) we conclude the

statement for A.

(vi) Any Reedy cofibrant A can be expressed as a transfinite compo-

410 Products

sition of the maps skm(A) → skm−1(A), so by (iv) and (v) we get the

statement of the theorem for any Reedy cofibrant A. This completes the

proof.

Recall from Corollary 16.2.5 and the remark at the beginning of the

proof of Proposition 19.2.2 above, for any cofibration f : U → V we

have the notation

srcΨ([k], f) = Σ([k];V ) ∪Σ([k];U) h([k];U)

and the map Ψ([k], f) goes from here to h([k];V ).

Corollary 19.2.4 Suppose A ∈ PC(M ) is Reedy cofibrant, k ∈ N,

and f : U → V is a cofibration in M . Then the map

A× srcΨ([k], f) → A× h([k]; v)


Proof In the cocartesian diagram

A× Σ([k];U) → A× h([k];U)

A× Σ([k];V )

↓

→ A× srcΨ([k], f)

↓

the upper arrow is a global weak equivalence inducing an isomorphism

on the set of objects, by the previous theorem. Furthermore it is a Reedy

cofibration by Proposition 15.6.12, so it is a Reedy isotrivial cofibration.

By Theorem 16.3.3, the bottom map is a global weak equivalence. The

statement of the corollary now follows by again using the previous The-

orem 19.2.3 as well as 3 for 2.

Theorem 19.2.5 Assume M is a tractable left proper cartesian model

category. For any A,B ∈ PC(M ), the map

A× B → Seg(A)× Seg(B)


Proof We suppose first that A and B are Reedy cofibrant. There is

a map A → A′ to an object satisfying the Segal conditions, which

is a transfinite composition of pushouts along morphisms of the form

srcΨ([k], f)Ψ([k],f)

→ h([k];V ). Each of these pushouts is a global trivial


Reedy cofibration inducing an isomorphism on the set of objects, by

Theorem 16.3.3. The map

A× B → A′ × B

is the corresponding transfinite composition of pushouts along mor-

phisms of the form

srcΨ([k], f)× B → h([k];U)× B.

This is because part of the cartesian hypothesis for M is (DCL) com-

mutation of direct products and colimits. By Corollary 19.2.4, the mor-

phisms srcΨ([k], f) × B → h([k];U) × B are global weak equivalences;

they are also Reedy cofibrations by Proposition 15.6.12 because we as-

sumed that B is Reedy cofibrant. These maps are again isomorphisms

on objects, so we can apply Theorem 16.3.3 which says that global triv-

ial cofibrations which induce isomorphisms on the set of objects are

preserved under pushout (and see Lemma 16.3.5 for the transfinite com-

position). Therefore A× B → A′ × B is a global weak equivalence.

Arguing in the same way for the product of a map B → B′ with A′,

then composing the two equivalences we conclude that the map

A× B → A′ × B′

is a global weak equivalence. On the other hand, A′ → Seg(A′) and

B′ → Seg(B′) are levelwise weak equivalences, so

A′ × B′ → Seg(A′)× Seg(B′)

is a levelwise weak equivalence. Similarly, the fact that A → A′ is a

global weak equivalence inducing an isomorphism on objects, implies

that

Seg(A) → Seg(A′)

is a levelwise weak equivalence, and by the same remark for B then

taking the product, we get that

Seg(A) × Seg(B) → Seg(A′)× Seg(B′)

is a levelwise weak equivalence. Thus we obtain a diagram

A× B → A′ × B′

Seg(A)× Seg(B)

↓

→ Seg(A′)× Seg(B′)

↓

412 Products

where the top map is a global weak equivalence, and the right vertical

and bottom maps are levelwise hence global weak equivalences (Lemma

14.5.3). By 3 for 2 it follows that the left vertical map is a global weak

equivalence as required for the theorem. This completes the proof for

the case of Reedy cofibrant objects.

Now suppose A and B are general objects of PC(M ). Consider Reedy

cofibrant replacements A′ → A and B′ → B; these may be chosen as

levelwise equivalences of diagrams, which are then global weak equiva-

lences by Lemma 14.5.3. In particular, Seg(A′) → Seg(A) is a levelwise

weak equivalence and the same for B′ so

Seg(A′)× Seg(B′) → Seg(A)× Seg(B)

is a levelwise weak equivalence. Note also that

A′ × B′ → A×B

is a levelwise weak equivalence of diagrams over ∆oOb(A)×Ob(B), so it is

a global weak equivalence by Lemma 14.5.3. The first part of the proof

treating the Reedy cofibrant case shows that

A′ × B′ → Seg(A′)× Seg(B′)

is a global weak equivalence. In the square diagram

A′ × B′ → A×B

Seg(A′)× Seg(B′)

↓

→ Seg(A) × Seg(B)

↓

the top, bottom and left vertical arrows are global weak equivalences,

so by 3 for 2 the right vertical arrow is a global weak equivalence. This

completes the proof.

Corollary 19.2.6 Suppose A → B and C → D are global weak equiv-

alences. Then the map

A× C → B ×D


Proof Suppose first of all that A, B, C and D are objects satisfying

the Segal conditions. Then the products also satisfy the Segal condi-

tions. Truncation of these is compatible with direct products, by Lemma

14.5.2, so the map in question is essentially surjective. By looking at the

19.3 The role of unitality, degeneracies and higher coherences 413

morphism objects we see that it is fully faithful, so it is a global weak

equivalence by following the definition.

Next suppose that A, B, C and D are any objects, and look at the

diagram

A× C → B ×D

Seg(A)× Seg(C)

↓

→ Seg(B)× Seg(D).

↓

The vertical maps are global weak equivalences by Theorem 19.2.5, while

the bottom map is a global weak equivalence by the first paragraph of

the proof. By 3 for 2, the top map is a global weak equivalence.

19.3 The role of unitality, degeneracies and higher

coherences

In this section, we point out why we need to impose the unitality condi-

tion A(x0) = ∗, to include the degeneracy maps in ∆ (which also corre-

spond to some sort of unit condition), and why we can’t truncate ∆ by,

say, dropping the objects [n] for n ≥ 4. These all have to do with the

arguments of this chapter about products. In some sense it goes back to

the Eilenberg-Zilber theorem; our product condition can be viewed as a

generalization to the present context where the information of direction

of arrows is retained.

19.3.1 The unitality condition

Suppose we tried to use non-unital precategories. These would be pairs

(X,A) where A : ∆oX →M is an arbitrary functor. The Segal condition

would include, for sequences of length n = 0, the fact that A(x0) → ∗

should be a weak equivalence, in other words A(x0) is weakly con-

tractible. So, this would constitute a weak version of the unitality con-

dition. We would proceed much as above, imposing the Segal condition

by the small object argument in an operation denoted A 7→ Segn(A).

It seems likely that this would lead to a model category, conjecturally

Quillen equivalent to the model categories on unital precategories which

we are constructing here.

However, even if the model structure existed, it could not be cartesian.

414 Products

The reason for this occurs at some very degenerate objects: consider the

non-unital precategory with object set Ob(B) = y a singleton, but

with functor the constant functor with values the initial object:

B(y, . . . , y) := ∅.

This includes the case of sequences of length 0: B(y) = ∅, so B doesn’t

satisfy the unitality condition. Now Segn(B) would be some kind of M -

enriched category with a single object; it seems clear that it would be

the coinitial ∗ but in any case has to contain ∗ as a retract.

Suppose A is another non-unital precategory (which might in fact be

unital). Consider A × B. The object set is Ob(A) × y ∼= Ob(A). But

for any sequence ((x0, y), . . . , (xn, y)) we have

(A× B)((x0, y), . . . , (xn, y)) = A(x0, . . . , xn)× B(y, . . . , y)

= A(x0, . . . , xn)× ∅ = ∅.

In particular, the structure of A×B depends only on Ob(A) and not on

A itself. This would be incompatible with the cartesian condition (for

any reasonable choice of M ), because

Segn(A) × ∗ → Seg

n(A)× Segn(B)

is contained as a retract, but Segn(A× B) is essentially trivial.

To make the last step of the above argument precise we would need to

investigate Segn explicitly. In the case M = Set, the same discussion

as in Section 16.8 applies, expressing Segn(A) as the category generated

by A considered as a system of generators and relations; the first step

would be to impose the Segal condition at n = 0 which, for M = Set,

is exactly the unitality condition; from there the rest is the same. In this

case we see that Segn(B) is really just ∗, Segn(A × B) is the discrete

category on object set Ob(A)× y, and it cannot contain Segn(A) as

a retract in general.

19.3.2 Degeneracies

Let Φ ⊂ ∆ denote the category consisting only of face maps, in other

words the objects of Φ are the nonempty finite linearly ordered sets [k]

for k ∈ N, but the maps are the injective order-preserving maps. We

could try to create a theory of weak categories based on Φ rather than

∆. It would be appropriate to call these “weak semicategories”, because

the degeneracy maps in ∆ correspond to inserting identity morphisms

19.3 The role of unitality, degeneracies and higher coherences 415

into a composable sequence. This theory is undoubtedly interesting and

important, and has not been fully worked out as far as I know.

This would undoubtedly be related to the work of J. Kock on weakly

unital higher categories [140] [141], as one could start by considering

weak semicategories, then impose a weak unitality condition.

Unfortunately the theory of products again doesn’t work if ∆ is re-

placed by Φ. Indeed, a slight modification of the example of the pre-

vious subsection again provides a counterexample. Let B be the pre-

category with a single object y, with B(y) = ∗ so it is unital, but with

B(y, . . . , y) = ∅ for any sequence of length n ≥ 1 (that is to say, with n+1

elements). This will still be a valid functor from Φy to M , however,

taking the product A× B will destroy the structure of A.

As in the previous subsection, this can be made precise in the case

M = Set. The non-unital precategories may then be considered as

systems of generators and relations for a category, but the system doesn’t

contain the degeneracies.

It is interesting to look more closely at how this works in the case of

systems of generators for a monoid, that is to say for a category with a

single object. The 1-cells of a precategory A correspond to generators of

the monoid, and the 2-cells correspond to relations of the form f = gh

among the generators. In this case, the system of generators and relations

is just given by two sets A(x, x) and A(x, x, x) with three maps

A(x, x, x) →→→ A(x, x).

The process of generators and relations would have to include the addi-

tion of identities.

In this case, for example, if A has a single generator and no relations,

then its product with itself A ×A will again be a system with a single

generator and no relations; but A generates the monoid N and N×N is

different from N, so the product of systems of generators and relations

doesn’t generate the product of the corresponding monoids.

One can see in this simple example how the degeneracies come to the

rescue. A system of generators and relations with unitality and degen-

eracies corresponds to a diagram of the form

A(x, x, x)→←→←→A(x, x) →←→ A(x) = ∗.

This means that there is an explicit element 1 among the generators,

with the relations 1 · f = f and f · 1 = f for any other generator f .

Now let’s look again at N generated by a system A consisting of a

416 Products

single generator. We have A(x, x) = f, 1 with relations corresponding

to the left and right identities for both f and 1 itself. The product now

has generators

A×A(x, x) = f × f, f × 1, 1× f, 1× 1

where we have noted the single object of A×A as x again rather than

(x, x). The unit generator is 1 × 1. The relations include the left and

right identities with the unit generator 1 × 1, plus two new relations of

the form

(f × 1) · (1× f) = f × f

and

(1× f) · (f × 1) = f × f.

The first of these two relations serves to eliminate the generator f×f so

we get to a monoid with two generators f ×1 and 1×f , then the second

relation gives the commutativity (f × 1) · (1× f) = (1× f) · (f × 1). So,

the monoid generated by the system A×A is indeed N× N.

Working out this example demonstrates how the degeneracies of ∆

enter into the cartesian condition in an important way.

19.4 Why we can’t truncate ∆

The above examples could all be done in the case of M = Set, where the

passage from precategories to categories is the process of generating a

category by generators and relations. For that, we didn’t need to consider

the part of ∆ involving [n] for n ≥ 4 (the case n = 3 being needed for

the associativity condition).

On the other hand, for weak enrichment in a general model category

M , we can’t replace ∆ by any finite truncation, that is by a subcategory

∆≤m of finite ordered sets of size ≤ m.

This can be seen by the requirement that there should be a higher

Poincare groupoid construction; in the case when M = K is the model

category of simplicial sets, the K -enriched precategories should be real-

izable into arbitrary homotopy types; and in particular the Segal groupoids

should be eqivalent to homotopy types by a pair of functors including

Poincare-Segal category and realization. These should be compatible

with homotopy groups in a way similar to that described in Chapters 3

and 4.

19.4 Why we can’t truncate ∆ 417

If we impose these conditions, it becomes easy to see that the Segal

groupoids defined using only ∆≤m (say for m ≥ 3) don’t model all

homotopy types. This means that, for the program we are pursuing

here, the category ∆≤m cannot be sufficient.

It should also be possible to show that ∆≤m can’t be used to model

homotopy n-types for n > m, in any way at all; however, it doesn’t seem

completely clear how to formulate a good statement of this kind.

If for 1-categories it suffices to look at ∆≤3, we could expect more

generally that in order to consider n-categories it would suffice to look

at ∆≤n+2, indeed this showed up in the explicit example of Chapter

17 and will show up again in our discussion of stabilization in a future

version (see [196]).

20

Intervals

Given our tractable, left proper and cartesian model category M , the

main remaining problem in order to construct the global model structure

on PC(M ) is to consider the notion of interval which should be an

M -precategory (to be called Ξ(N |N ′) in our notations below), weak

equivalent to the usual category E with two isomorphic objects υ0, υ1 ∈

E , and with a single morphism between any pair of objects.

If A ∈ PC(M ) is a weakly M -enriched category, an internal equiv-

alence between x0, x1 ∈ Ob(A) is a “morphism from x0 to x1” (see

(20.2.1) below), which projects to an isomorphism in the truncated cat-

egory τ≤1(A). This terminology was introduced by Tamsamani in [206].

It plays a vital role in the study of global weak equivalences. Essen-

tial surjectivity of a morphism f : A → B means (assuming that B

is levelwise fibrant) that for any object y ∈ Ob(B), there is an object

x ∈ Ob(A) and an internal equivalence between f(x) and y.

Unfortunately, an internal equivalence between x0 and x1 in A doesn’t

necessarily translate into the existence of a morphism E → A. This

will work after we have established the model structure on PC(M ) if

we assume that A is a fibrant object. However, in order to finish the

construction of the model structure, we should start with the weaker

hypothesis1 that A satisfies the Segal conditions and is levelwise fibrant.

The “interval object” Ξ(N |N ′) should be contractible, and have the

versality property that whenever x0 and x1 are internally equivalent,

there is a morphism Ξ(N |N ′) → A relating them.

The construction of such a versal interval was the subject of an error

in [193], found and corrected by Pelissier in [171]. This was somewhat

similar to a mistake in Dwyer-Hirschhorn-Kan’s original construction of

1 As observed by Bergner [36] this hypothesis will be equivalent to fibrancy in theglobal projective model structure, once we know that it exists.


20.1 Contractible objects and intervals in M 419

the model category structure for simplicial categories ([87], now [88]),

pointed out by Toen and subsequently fixed for simplicial categories by

Bergner [33]. Pelissier fixed this problem for the model category of Segal

categories by constructing an explicit interval object and verifying its

topological properties using the comparison between Segal 1-groupoids

and spaces. Drinfeld has constructed an interval object for differential

graded categories [83].

Pelissier’s correction as written covered only the case of K -enriched

weak categories, and one of our purposes here is to point out that his

argument serves to construct the required intervals in general, by func-

toriality with respect to a left Quillen functor K →M . For the main

result which is contractibility of the Ξ(N |N ′) we proceed therefore in

two steps: first considering the problem for the case of Segal categories

i.e. K -enriched weak categories as was done in [171]; then going to the

case of M -enriched weak categories by transfer along K → M . Sec-

tions 11.8, 14.7, and 16.7 about transfer along a left Quillen functor were

motivated by this movement. The possibility of doing that is one of the

advantages of the fully iterative point of view originally suggested by

Andre Hirschowitz in Pelissier’s thesis topic, in which M is a general

input into the construction. It should also be possible to adapt Pelissier’s

correction directly to the original n-nerves considered in [206] [193], by

using Tamsamani’s theorems on the topological realization of weak n-

groupoids which in turn applied Segal’s original results in a partially

iterative way. That would be more geometrically motivated, but for the

present treatment the fully iterative approach is both more general and

more direct.

I would like to thank Regis Pelissier for finding and correcting this

problem.

20.1 Contractible objects and intervals in M

An object A ∈ M is contractible if the unique morphism A → ∗ is a

weak equivalence. An interval object is a triple (B, i0, i1) where B ∈M

and i0, i1 : ∗ → B such that B is contractible and i0 ⊔ i1 : ∗ ∪∅ ∗ → B

is a cofibration.

Assumption (AST) in the cartesian condition 10.0.9 says that ∗ is a

cofibrant object, so an interval object is itself cofibrant.

A morphism between intervals from (B, i0, i1) to (B′, i′0, i′1) means a

420 Intervals

morphism f : B → B′ such that f i0 = i′0 and f i1 = i′1. Since B and

B′ are contractible, a morphism f is automatically a weak equivalence.

Lemma 20.1.1 Suppose (B, i0, i1) and (B′, i′0, i′1) are two interval ob-

jects. Then there is a third one (B′′, i′′0 , i′′1) and morphisms of intervals

f : B → B′′ and f ′ : B′ → B′′. These may be assumed to be trivial

cofibrations.

Proof Put A := B ∪∗∪∅∗ B′ and choose a factorization

Af→ B′′ → ∗

where the first morphism is a cofibration and the second morphism is a

weak equivalence. Now i0 and i′0 are the same when considered as maps

∗ → A because of the coproduct in the definition of A. Thus f i0 = f i′0gives a map i′′0 : ∗ → B′′. Similarly i′′1 := f i1 = f i′1. The map

∗ ∪∅ ∗ → A is cofibrant, and since f is cofibrant the composition into

B′′ is cofibrant. Note that the mapsB → A and B′ → A are cofibrations,

so the same is true of the maps to B′′, and since the intervals are weakly

equivalent to ∗ these maps are trivial cofibrations.

Recall that we defined in Chapter 14 a functor τ≤0 : M → Set by

τ≤0(A) := Homho(M )(∗, A′) where A → A′ is a fibrant replacement.

Lemma 20.1.2 Suppose A is a fibrant object and a, b : ∗ → A. The

following conditions are equivalent:

(a)—The classes of a and b in τ≤0(A) coincide;

(b)—for any interval object (B, i0, i1) there exists a map B → A sending

i0 to a and i1 to b;

(c)—there exists an interval object (B, i0, i1) and a map B → A sending

i0 to a and i1 to b.

Proof This is an exercise in Quillen’s theory of the homotopy category

of a model category, which we do for the reader’s convenience.

Note that (b) ⇒ (c) ⇒ (a) easily. Assume that A is also cofibrant.

To prove that (a) ⇒ (c), suppose that the classes of a and b coincide

in τ≤0(A). This is equivalent to saying that the two maps a, b : ∗ → A

project to the same map in ho(M ). Recall from Quillen [175] that ho(M )

is also the category of fibrant and cofibrant objects of M , with homo-

topy classes of maps. As ∗ is automatically fibrant, and cofibrant by

hypothesis; and we are assuming that A is cofibrant and fibrant, then

condition (a) says that the two maps a and b are homotopic in the sense

of Quillen [175], which says exactly condition (c). For the implication


(a) ⇒ (c), but with A assumed only to be fibrant, choose a trivial fi-

bration from a cofibrant object A′ → A. Lift to maps a′, b′ : ∗ → A′.

Since A′ → A projects to an isomorphism in ho(M ), the maps a′ and

b′ are equivalent in τ≤0(A′) so by condition (c) proven for A′ previously,

there exists an interval object (B, i0, i1) and an extension of a′ ⊔ b′ to

B → A′. Composing gives the required map B → A.

To finish the proof it suffices to show (c) ⇒ (b). Suppose (B, i0, i1)

and (B′, i′0, i′1) are two interval objects. Applying Lemma 20.1.1 there is

an interval object (B′′, i′′0 , i′′1) with trivial cofibrations from both B and

B′. If a ⊔ b : ∗ ∪∗ ∗ → A extends to a map B → A, and if A is fibrant,

then the lifting property for A gives the extension to B′′ → A which

then restricts to a map B′ → A as required to show (c)⇒ (b).

Using the assumption that M is cartesian, we can make a similar

statement explaining the relation of homotopy between morphisms using

an interval, if the target is a fibrant object of M .

Lemma 20.1.3 Suppose M is a cartesian model category. Suppose A

is a cofibrant object and C is a fibrant object. Then, for two morphisms

f, g : A → B the following statements are equivalent:

(a)—f and g are homotopic in Quillen’s sense, meaning that the classes

of f and g in Homho(M )(A,C) coincide;

(b)—for any interval object (B, i0, i1) there exists a map h : A×B → C

such that h (1A × i0) = f and h (1A × i1) = g;

(c)—there exists an interval object (B, i0, i1) and a map h : A×B → C

such that h (1A × i0) = f and h (1A × i1) = g.

Proof The cartesian property of M implies that for any interval object

B, the diagram

A× (∗ ∪∅ ∗) = A ∪∅ A → A× B → A

is an A × I-object in Quillen’s sense, and so can be used to measure

homotopy between our two maps.

20.2 Intervals for M -enriched precategories

Let E := Υ(∗) denote the category with two objects υ0, υ1 and a single

morphism between them. Thus, E(υ0, . . . , υ0) = ∗, E(υ1, . . . , υ1) = ∗,

E(υ0, . . . , υ0, υ1, . . . , υ1) = ∗ and the remaining values are ∅. This is the

image of the usual category [0 → 1] under the mapPC(Set) → PC(M ).

422 Intervals

An alternative description of E in terms of the representable object

notation of Section 12.5 is E = h([1], ∗). If A is any M -enriched pre-

category, a map E → A is the same thing as a triple (x0, x1, a) where

x0, x1 ∈ Ob(A) and a : ∗ → A(x0, x1) is an element of the “set of

morphisms from x0 to x1”. This “set of morphisms” may be denoted by

Mor1A(x0, x1) := HomM (∗,A(x0, x1)) = Homx0,x1

PC(M )(E ,A) (20.2.1)

where the superscript on the right designates the subset of maps E → A

sending υ0 to x0 and υ1 to x1.

Let E denote the image of the category with two isomorphic objects

under the map PC(Set) → PC(M ). We think of E as containing E

as a subcategory. Thus E again has objects υ0, υ1, but E(x0, . . . , xp) =

∗ for any sequence of objects. One can also view it as the codiscrete

precategory with two objects, E = codsc([1]) = codsc(υ0, υ1) in the

notation of Section 12.5.

If A ∈ PC(M ), a map E → A is sure to correspond to an internal

equivalence between the images of the two endpoints υ0, υ1. Say that

a map E → A which extends to E → A, is strongly invertible. An

important little observation is that the identity morphisms (i.e. those

given by the images of the degeneracies ∗ = A(x0) → A(x0, x0)) are

strongly invertible.

Unfortunately, given a general morphism from x0 to x1 in A, the

corresponding map E → A will not in general extend to E → A. That

is to say, not all internal equivalences will be strongly invertible. This

is why we need to do some further work to construct versal interval

objects.

Assuming that A satisfies the Segal condition and is levelwise fibrant,

suppose x0, x1 ∈ Ob(A) and suppose a : ∗ → A(x0, x1) is a morphism

from x0 to x1. The condition of a being an internal equivalence means

that there should be morphisms b and c from x1 to x0, such that ba is

homotopic to the identity of x0 and ac is homotopic to the identity of

x1. In turn, these homotopies can be represented by maps from interval

objects in M which we shall denote by N and N ′ respectively. We will

build up a big coproduct representing this collection of data.

It turns out to be convenient to relax slightly the conditions that the

homotopies go between ab and the identity (resp. ca and the identity).

Instead, we say that the homotopies go between ab or ca and strongly

invertible morphisms. In particular, the source of c could be an object


x′1 different from x1 and the target of b could be an object x′0 different

from x0.

This situation can be represented diagramatically by

x′1

x1

x0

x′0

⇔N ′

⇔N

c

a

b

@@@@@@R

U

K

U

.

The goal in this section is to construct a precategory Ξ(N |N ′) such that

a map Ξ(N |N ′) → A is the same thing as such a diagram. Notice that

the diagram may be divided into two triangles which are independent

except for the fact that they share the same edge labeled a. Our Ξ(N |N ′)

will be a coproduct of two precategories Ξ(N) and Ξ(N ′) along E , where

each of the pieces represents a triangular diagram.

So to start, suppose given an interval object (N, i0, i1) with N ∈ M

and i0, i1 : ∗ → N . We will construct a precategory Ξ(N) such that a

map from Ξ(N) to A is the same thing as a diagram of the form

y1

y0

y2

⇔N

QQQQ

QQs

+U

in A. There are three pieces. The part involvingN is a map to A from an

M -enriched precategory of the form Υ(N) (see Section 16.1 and Chap-

ter 18), which comes with two maps Υ(i0) and Υ(i1) from E to Υ(N).

The commutative triangle corresponds to a map from a representable

precategory of the form h([2], ∗) to A. The strongly invertible morphism

424 Intervals

on the left corresponds to an extension of one of the E → Υ(N) → A

to a map E → A.

Motivated by this picture, define the M -enriched precategory Ξ(N)

to be the coproduct of three terms corresponding to these three pieces:

Ξ(N) := E ∪Υ(i0)(E) Υ(N) ∪Υ(i1)(E) h([2], ∗).

The map at the end of the coproduct notation is E = h([1], ∗) → h([2], ∗)

corresponding to the edge [1] → [2] sending 0 to 0 and 1 to 2. The objects

of Ξ(N) will be denoted ξ0, ξ1, ξ2. These correspond to the three objects

of h([2], ∗). In case of a map Ξ(N) → A corresponding to a diagram as

above, the images of ξi are the objects labeled yi above. Thus the two

objects υ0, υ1 of both copies of E as well as Υ(N) correspond to ξ0 and

ξ2 respectively.

Lemma 20.2.1 Suppose A ∈ PC(M ). Then a map Ξ(N) → A cor-

responds to giving three objects x0, x1, x2 ∈ Ob(A), to giving an element

t : ∗ → A(x0, x1, x2), to giving a map b : N → A(x0, x2) and to giving

a map g : E → A such that b Υ(i1) = ∂02(t), and b Υ(i0) = g(e01)

where e01 : ∗ → E(υ0, υ1) is the unique map.

Proof This comes from the coproduct description for Ξ(N).

We think of t : ∗ → A(x0, x1, x2) as corresponding to a commutative

triangle with maps ∂01(t) and ∂12(t) whose “composition” is

∂12(t) ∂01(t) = ∂02(t).

Then N can be a homotopy from ∂02(t) to the map g(e01) (in our ap-

plication N will be contractible). Then the extension of this map to g

defined on E says that g(e01) is strictly invertible. So, roughly speaking

when we look at a map Ξ(N) → A we are looking at two morphisms

whose composition ∂12(t) ∂01(t) is equivalent to an invertible map.

The two different maps in question correspond to maps ζ01, ζ12 :

E → Ξ(N) with ζ01 corresponding to ∂01(t) and ζ12 to ∂12(t).

The construction Ξ also works for the other half of Ξ(N |N ′). We

distinguish the two interval objects which are used here, for clarity of

notation. Obviously one could choose the same on both sides.

Given two interval objects N and N ′, we can form

Ξ(N |N ′) := Ξ(N) ∪E Ξ(N ′)

where the map E → Ξ(N) is ζ01 and the map E → Ξ(N ′) is ζ12. These

become the same map denoted η : E → Ξ(N |N ′). Denote the four


objects of Ξ(N |N ′) by ξ|0, ξ0|1, ξ1|2 and ξ2|, these corresponding with

the objects of Ξ(N) or Ξ(N ′) by saying that ξi|j corresponds to ξi in the

left piece Ξ(N) and to ξj in the right piece Ξ(N ′) to give the following

picture of Ξ(N |N ′):

ξ|0

ξ1|2

ξ0|1

ξ2|

Υ(N ′)

Υ(N)

EE

h([2], ∗)

h([2], ∗)

η

@@@@

@@R

U

K

U

displaying η as a morphism from ξ0|1 to ξ1|2 i.e. an element of the set

Mor1Ξ(N |N ′)(ξ0|1, ξ1|2) defined in (20.2.1).

Lemma 20.2.2 Suppose (N, i0, i1) and (N ′, i′0, i′1) are interval objects

of M . If A is an M -enriched precategory, then a map Ξ(N |N ′) → A

corresponds to the data of a morphism (x0, x1, a) in A, of two other

objects x′0 and x′1, together with commutative triangles

s : ∗ → A(x0, x1, x′0), t : ∗ → A(x′1, x0, x1),

with maps h : Υ(N) → A and h′ : Υ(N ′) → A and two maps u, v :

E → A such that various maps E → A induced by these data coincide

(see the diagram in the proof below).

Proof This comes from the coproduct description of Ξ(N |N ′) and the

corresponding properties for Ξ(N) and Ξ(N ′). The objects x0 and x1are the images of ξ0|1 and ξ1|2 while x′0 is the image of ξ2| and x

′1 is the

image of ξ|0.

The maps which are supposed to coincide may be read off from the

426 Intervals

diagram

x′1

x1

x0

x′0

⇔h′

⇔h

t

a

s

u

v

@@@@

@@R

U

K

U

in which the 2-cells represent maps from Υ(N) or Υ(N ′), the thick lines

represent maps from E , and the triangles represent maps from h([2], ∗).

For example the boundary ∂02 t : ∗ → A(x′1, x1) corresponds to a map

E → A sending υ0 to x′1 and υ1 to x1. This should be the same as the

map

E = Υ(∗)Υ(i′0)→ Υ(N ′)

h′

→ A.

The other identifications are similar.

Lemma 20.2.3 For any two interval objects N and N ′, the M -precategory

Ξ(N |N ′) is Reedy cofibrant, and indeed the inclusion

discξ0|1, ξ1|2 → Ξ(N |N ′)

is Reedy cofibrant hence injectively cofibrant.

Proof Use Corollaries 15.6.5 and 15.6.6, and Lemma 16.1.3.

However, Ξ(N |N ′) is not projectively cofibrant, because the inclusions

of edges E → h([2], N) are Reedy but not projective cofibrations. This

issue will be addressed further in the comments after Remark 20.3.2

below.

Record here what happens when we change the intervals used in the

construction.

Lemma 20.2.4 Suppose f : N → P is a morphism between interval

objects (N, i0, i1) and (P, j0, j1), that is f i0 = j0, f i1 = j1. Suppose

similarly f ′ : N ′ → P ′ is a morphism between interval objects (N ′, i′0, i′1)

and (P ′, j′0, j′1). Then these induce a global weak equivalence

Ξ(N |N ′) → Ξ(P |P ′).

20.3 The versality property 427

Proof It is a levelwise weak equivalnce, being a pushout of maps which

are levelwise a weak equivalences.

20.3 The versality property

From the universal property of Lemma 20.2.2, we obtain the versality

property of Ξ(N |N ′).

Theorem 20.3.1 Suppose A ∈ PC(X,M ) satisfies the Segal condi-

tion and is fibrant in the Reedy diagram model structure FuncReedy(∆oX/X,M ).

Suppose that x, y ∈ X = Ob(A) and a : ∗ → A(x, y) is an element

of Mor1A(x, y). Suppose that a is an inner equivalence, in other words

the image of a in the truncated category τ≤1(A) ∈ Cat, is invertible.

Then for any interval objects N and N ′ in M there exists a morphism

Ξ(N |N ′) → A sending ξ0|1 and ξ2| to x and ξ1|2 and ξ|0 to y, sending

the tautological morphism η to a, and sending the two copies of E to the

identities of x and y respectively.

Proof Since A satisfies the Segal condition, the truncation τ≤1(A) may

be defined using A itself, that is to say that the truncation is the 1-

category with x + Ob(A) as set of objects, and whose nerve relative to

this set is the functor

∆ox → Set, (x0, . . . , xn) 7→ τ≤0A(x0, . . . , xn).

The Reedy fibrant condition for A implies that it is levelwise fibrant,

which means that for any sequence (x0, . . . , xn) ∈ ∆oX the imageA(x0, . . . , xn)

is a fibrant object of M , so

τ≤0A(x0, . . . , xn) = HomM (∗,A(x0, . . . , xn))/ ∼

where ∼ is the relation of homotopy occuring in Lemma 20.1.2.

The fact that a maps to an isomorphism in τ≤1A therefore means that

there is an inverse b ∈ τ≤0A(y, x); and by the levelwise fibrant condition

it can be represented by b : ∗ → A(y, x).

By the Segal condition the morphism

A(x, y, x) → A(x, y) ×A(y, x)

is a weak equivalence. On the other hand, the Reedy fibrant condition

in the diagram category means that the matching map at (x, y, z) is a

fibration in M , which in turn implies that the Segal map above is a

428 Intervals

fibration. Hence it is a trivial fibration, in particular the element (a, b) :

∗ → A(x, y)×A(y, x) lifts to a map ∗ → A(x, y, x). This gives a diagram

s : h([2], ∗) → A

representing “the composition b a”, fitting into the lower left triangle

in the picture on page 426. The image of s in the nerve of τ≤1A is the

commutative triangle for the composition of the images of b and a. We

chose b as representing an inverse to a in the truncated category, so the

02 edge of s is homotopic to the identity of x; by Lemma 20.1.2 there

exists a map N → A(x, x) representing this homotopy, or by adjunction

h : Υ(N) → A.

This gives the lower or Ξ(N) part of the required diagram. A similar

discussion using the fact that ab is homotopic to the identity of y, gives

the upper or Ξ(N ′) part, and these glue together to give the required

map Ξ(N |N ′) → A.

Remark 20.3.2 Let Ξproj(N |N ′) denote a cofibrant replacement for

Ξ(N |N ′) in the projective diagram category FuncReedy(∆oX/X,M ). Then

it has the same versality property with respect to any A which is levelwise

fibrant and satisfies the Segal condition.

The projectively cofibrant version Ξproj(N |N ′) could be constructed

explicitly by inserting objects of the form Υ(L) and Υ(L′) in between ζ

and the two triangles, for intervals L and L′, according to the picture

ξ|0

ξ1|2

ξ0|1

ξ2|

N ′

NLL′

@@@@@@R

U

K

U

.

This corresponds to the step in the proof of the theorem where we used

the Reedy fibrant property to lift (a, b) to an element s : ∗ → A(x, y, x).

If A is assumed only fibrant in the projective structure (i.e. levelwise

fibrant) then (a, b) only lifts up to a homotopy given by L → A(x, y)×

20.4 Contractibility of intervals for K -precategories 429

A(y, x). The second term may be neglected since we don’t care about the

bottom arrow of the big diagram, and the piece L → A(x, y) corresponds

to a map Υ(L) → A. We presented the Reedy version in our main

discussion above because the diagrams are easier to picture.

Let Ξ(N |N ′) ⊂ Seg(Ξ(N |N ′)) be the full subcategory containing only

the two objects ξ|0 and ξ2|. In the situation of the theorem, we get by

restriction plus functoriality of Seg a map

Ξ(N |N ′) → Seg(A).

Similarly if Ξproj(N |N ′) ⊂ Seg(Ξproj(N |N ′)) is the full subcategory

containing only ξ|0 and ξ2|, then in the situation of the remark we get a

map as stated in the following corollary.

Corollary 20.3.3 Suppose A is levelwise fibrant and satisfies the Se-

gal conditions, and suppose x, y ∈ Ob(A) are two internally equivalent

objects. Then there is a map

Ξproj(N |N ′) → Seg(A)

sending the two objects of Ξproj(N |N ′) to x and y respectively.

Proof As in the above remark we get a map Ξproj(N |N ′) → A, hence

by functoriality of Seg and composition with the inclusion,

Ξproj(N |N ′) ⊂ Seg(Ξproj(N |N ′)) → Seg(A)

as required.

It remains to be seen that Ξ(N |N ′)) and thus Ξ(N |N ′) are con-

tractible.

20.4 Contractibility of intervals for K -precategories

Given the above construction, the main problem is to prove that Ξ(N |N ′)

is contractible. In this section we do that for enrichment over the Kan-

Quillen model category K of simplicial sets.

Theorem 20.4.1 Suppose N, i0, i1 and N ′, i′0, i′1 are two interval ob-

jects in the Kan-Quillen model category of simplicial sets K . Then

Ξ(N |N ′) is contractible in PC(K ), that is Ξ(N |N ′) → ∗ is a global

weak equivalence. We have a map Ξ(N |N ′) → E × E which is a global

430 Intervals

weak equivalence and an isomorphism on the sets of objects. In particular

the map

Seg(Ξ(N |N ′)) → E × E

induces an objectwise weak equivalence, which is to say that

Seg(Ξ(N |N ′))(x0, . . . , xp) is contractible in M

for any sequence of objects x0, . . . , xp ∈ ξ|0, ξ0|1, ξ1|2, ξ2|.

Proof This was treated in the last chapter of [171] and our present

version is only slightly different in that we have expanded somewhat Ξ

as something with 4 objects. Our present picture is perhaps closer to

Drinfeld’s intervals for DG-categories [83].

Elements of PC(K ) may be considered as certain kinds of bisimpli-

cial sets (see Section 12.7 and Chapter 17), and this commutes with

coproducts. Similarly the diagonal realization from bisimplicial sets to

simplicial sets commutes with coproducts and takes Reedy or injec-

tive cofibrations2 in PC(K ) to cofibrations in K (which are just the

monomorphisms). Call the composition of these two operations | · | :

PC(K ) → K . Note that |E|, |E|, and |h([2], ∗)| are contractible sim-

plicial sets, and if N is an interval object in K then |Υ(N)| is con-

tractible. Thus, |Ξ(N)| is a successive cofibrant coproduct of contractible

objects over contractible objects, so it is contractible. Similarly the co-

product of two of these over the contractible |E| (mapping into both

sides by cofibrations) is contractible, so |Ξ(N |N ′)| is contractible. In

general a map A → Seg(A) induces a weak equivalence of simplicial

sets |A|∼→ |Seg(A)|. Thus in our case, |Seg(Ξ(N |N ′)| is contractible.

On the other hand, all of the 1-morphisms in Ξ(N |N ′) go to invert-

ible morphisms in Seg(Ξ(N |N ′), in effect the main middle morphism

η has by construction a left and a right inverse up to equivalence; so

it goes to an equivalence, and its inverses go to equivalences too. Thus

Seg(Ξ(N |N ′) is a Segal groupoid. Now, a Segal groupoid whose re-

alization is contractible, is contractible (see Proposition 17.2.3). Thus

Seg(Ξ(N |N ′) is contractible, which proves the theorem in the case of

K .

2 The Reedy and injective cofibrations are the same in PC(K ) by Proposition15.7.2 as was pointed out in Corollary 17.0.2.

20.5 Construction of a left Quillen functor K →M 431

20.5 Construction of a left Quillen functor K →M

In order to transfer the above contractibility result for Ξ(N |N ′) in the

K -enriched case, to the general case, we explain in this section the

essentially well-known construction of a left Quillen functor K →M .

In Hovey [120] was explained the intuition that every monoidal model

category is a module over K , and even without the monoidal structure

there is a left Quillen functor from K into M . The construction is based

on a choice of contractible cosimplicial object in M , or more precisely

a cosimplicial resolution in the sense of Hirschhorn [116]. That means a

functor R : ∆ →M which is cofibrant in the Reedy model structure

FuncReedy(∆,M ).

Recall that an object A ∈ M is contractible if the unique morphism

A → ∗ is a weak equivalence. We say that a cosimplicial object R :

∆ →M is levelwise contractible if R([n]) is contractible for each object

[n] ∈ Ob(∆).

Lemma 20.5.1 There exists a choice of Reedy-cofibrant levelwise con-

tractible cosimplicial object R : ∆ →M .

Proof See [116].

We fix one such choice, from now on. The objects R([n]) may be

thought of as the “standard n-simplices” in M . If A ∈M , defineR∗(A) :

∆o → Set to be the functor

R∗(A) : [n] 7→ HomM (R([n]), A).

Theorem 20.5.2 If R is a Reedy-cofibrant levelwise contractible cosim-

plicial object, then R∗ is a right Quillen functor from M to K =

Func(∆o,Set). Its left adjoint

R! : K →M

is a left Quillen functor given by the usual formula for the topological

realization of a simplicial set, but with the standard n-simplex replaced

by R([n]) ∈M .

Proof See [116].

Corollary 20.5.3 The realization functor induces, for every object set

X, a functor

PC(X ;R!) : PCc(X,K ) → PCc(X,M ).

This is a left Quillen functor for c denoting either the projective, injective

432 Intervals

or Reedy model structures on K and M -enriched precategories over X.

It is compatible with change of set X, and gives a functor

PC(R!) : PC(K ) → PC(M )

from the Segal precategories to the M -enriched precategories.

Proof Combine Theorem 20.5.2, with the discussion of Proposition

14.7.2.

The corresponding right Quillen functorPC(R∗) : PC(M ) → PC(K )

should be applied to A ∈ PC(M ) only after taking a fibrant replace-

ment A → A′. Define IntR(A) := PC(R∗)(A′). We call this the R-

interior of A, since it is obtained by looking at maps from the standard

simplices R([n]) into A(x0, . . . , xn) so it measures A “from the inside”.

This construction is compatible with truncation:

Lemma 20.5.4 For A ∈ PC(M ) we have an isomorphism of cate-

gories τ≤1(A) ∼= τ≤1(IntR(A)).

Proof This follows from the definition of τ≤1 in Section 14.5.

By its construction as a colimit, PC(R!) preserves coproducts, pre-

serves constructions Υ and h, preserves the various notions of cofibrancy.

Since M is cartesian, Proposition 16.7.3 says that PC(R!) takes weak

equivalences in PC(X,K ) to weak equivalenes in PC(X,M ). Further-

more since it preserves truncations,PC(R!) preserves global weak equiv-

alences, and preserves the notion of interval objects.

20.6 Contractibility in general

We can now use the functorPC(X ;R!) to transfer the the contractibility

result for K -enriched precategories, to PC(M ). This yields the first

main theorem of the present chapter saying that Ξ(N |N ′) is a good

interval object in PC(M ). This was the last step missing in Pelissier’s

[171] correction of [193], but which is actually straightforward from a

Quillen-functorial point of view.

The contractibility statement is made before we have completely fin-

ished the construction of the model structure, although it is the penulti-

mate step. Some care is still therefore necessary in using only the parts

of the model structure which are already known.

20.6 Contractibility in general 433

Theorem 20.6.1 Suppose N, i0, i1 and N ′, i′0, i′1 are two interval ob-

jects. Then Ξ(N |N ′) is contractible in PC(M ), that is Ξ(N |N ′) → ∗

is a global weak equivalence. We have a map Ξ(N |N ′) → E × E which

is a global weak equivalence and an isomorphism on the sets of objects.

In particular the map

Seg(Ξ(N |N ′)) → E × E

induces an objectwise weak equivalence, which is to say that

Seg(Ξ(N |N ′))(x0, . . . , xp) is contractible in M

for any sequence of objects x0, . . . , xp ∈ ξ|0, ξ0|1, ξ1|2, ξ2|.

Proof First notice that the statement of the theorem is independent of

the choice of interval object: if N → P andN ′ → P ′ are maps of interval

objects then the statement of the theorem for (N,N ′) is equivalent to

the statement for (P, P ′). See Lemma 20.2.4. In particular it suffices to

prove the statement for one pair of intervals.

Theorem 20.4.1 gives the same statement for precategories enriched

over the Kan-Quillen model category K of simplicial sets. Then choose

a left Quillen functor R! : K →M . This gives a functor PC(R!) :

PC(K ) → PC(M ) which preserves coproducts. Suppose (B, i0, i1) is

an interval object in K . Then R!(B) is an interval object in M and

PC(R!)(Ξ(B|B) = Ξ(R!(B)|R!(B)).

Now since PC(R!) preserves global weak equivalences, we obtain the

statement of the theorem for the pair of interval objects R!(B)|R!(B).

By the invariance discussed in the first paragraph of the proof, this

implies the statement of the theorem for all N,N ′.

Recall that Ξ(N |N ′) ⊂ Seg(Ξ(N |N ′)) is the full subcategory contain-

ing only the two objects ξ|0 and ξ2|. Since all objects of Seg(Ξ(N |N ′))

are equivalent, the inclusion

Ξ(N |N ′) → Seg(Ξ(N |N ′))

is a global weak equivalence (it is by definition fully faithful and both

sides satisfy the Segal conditions). It follows from Theorem 20.6.1 and

the 3 for 2 property of global weak equivalences, that the functor

pN,N ′ : Ξ(N |N ′) → E

is a global weak equivalence; furthermore this induces isomorphisms on

the sets of objects (there are exactly two objects on each side), and

434 Intervals

both sides satisfy the Segal conditions, so pN,N ′ is an objectwise weak

equivalence of diagrams.

20.7 Pushout of trivial cofibrations

These interval objects allow us to analyze pushouts along trivial cofibra-

tions which are not isomorphisms on objects. In this discussion, we use

injective cofibrations since this encompasses the Reedy and projective

cofibrations too.

We start by considering the pushout along the standard interval E .

Lemma 20.7.1 Suppose A ∈ PC(M ), and y ∈ Ob(A). Then the

pushout morphism

a : A → A∪y E

obtained by identifying υ0 and y, is a global weak equivalence.

Proof By Corollary 19.2.6 applied to the identity of A and the map

p : E → ∗, the map

1A × p : A× E → A

is a global weak equivalence. Let i0, i1 : ∗ → E be the two inclusions of

objects υ0 and υ1. The two maps

1A × i0, 1A × i1 : A → A× E

are global weak equivalences, as can be seen by composing with 1A × p

and using 3 for 2.

Now, consider the morphism g : E × E → E × E equal to the identity

on E × υ0 and sending E × υ1 to the single object (υ0, υ1). Set

B := A∪y E .

Let q : B → A denote the projection obtained by sending all of E to the

single object y ∈ Ob(A). Then

B × E =(A× E

)∪υ0×E

(E × E

).

Apply the map g to the second factor of this pushout, to obtain a map

f : B × E → B × E

such that f restricts to the identity on B × υ0, while f |B×υ1 is the


projection q : B → A. By the first paragraph of the proof, the maps

induced by f ,

B × υ0 → B × E

and

B × υ1 → B × E

are global weak equivalences. The composition of 1B × p : B × E → B

with the morphism f considered above, is a morphism

(1B × p) f : B × E → B

such that the composition (1B × p) f (1B × i0) is the identity of B,

and the composition (1B × p) f (1B × i0) is the composition

Bq→ A

a→ B.

The facts that (1B × p) f (1B × i0) is the identity of B, and that

(1B× i0) is a global weak equivalence, imply by 3 for 2 that (1B× p) f

is a global weak equivalence. But then, composing with the global weak

equivalence (1B× i1) we see that (1B× p) f (1B× i0) is a global weak

equivalence, in other words that the composition aq is a global weak

equivalence. In the other direction, the composition

Aa→ B

q→ A

is the identity ofA. Thus, we conclude from the last sentence of Theorem

14.6.4 that both q and the inclusionA → B are global weak equivalences.

This last statement is what we are supposed to prove.

Corollary 20.7.2 Suppose B is an M -enriched precategory with two

objects b0, b1. Suppose B satisfies the Segal conditions, and is contractible,

that is the map B → ∗ is a global weak equivalence. Then for any

A ∈ PC(M ) and any object y ∈ Ob(A) the map

A → A∪b0 B

obtained by identifying b0 to y, is a global weak equivalence.

Proof There is a unique map f : B → E sending b0 to υ0 and b1 to υ1.

This map is a global weak equivalence, as seen by applying 3 for 2 to

the composition

B → E → ∗ .

But since it induces an isomorphism on objects, and both sides satisfy

436 Intervals

the Segal conditions, it is an objectwise weak equivalence of diagrams.

Applying f to the second piece of the given pushout, we get a map

g : A ∪b0 B → A ∪y E

to the pushout considered in the previous corollary. However, g is an

objectwise weak equivalence of diagrams, so it is a global weak equiva-

lence. Note that g commutes with the maps from A, so by the previous

corollary and 3 for 2 we conclude that the map of the present statement


Suppose f : A → B is an injective trivial cofibration, and suppose B

is levelwise fibrant and satisfies the Segal conditions. Let Z := Ob(B)−

f(Ob(A)). For each z ∈ Z choose e(z) ∈ Ob(A) and

a(z) ∈ B(f(e(z)), z)

such that the image of a(z) is invertible in the truncated category. This

is possible by the definition of essential surjectivity of A → B.

Applying Theorem 20.3.1 There exist collections of interval objects

Nz, N′z indexed by z ∈ Z, and functors ti : Ξ(Nz |N ′

x) → B sending ξ|0to e(z), ξ2| to z, and sending the tautological morphism η to a(z). By

functoriality of the construction Seg we get

Seg(Ξ(Nz |N′x)) → Seg(B),

and restricting this gives ti : Ξ(Nz |N′z) → Seg(B). Now, ti sends the

first object to e(z) ∈ A and the second object to z. Putting these all

together we get a morphism in PC(M ),

A ∪∐

z∈Ztiξ(|0)∐

z∈Z

Ξ(Nz|N′z)

T→ Seg(B),

and now T induces an isomorphism on sets of objects. It is no longer

necessarily a cofibration. We would like to show that T is a global

weak equivalence. We start by considering pushouts along the inter-

val 1-category E . For this proof we make essential use of the cartesian

property of M and the discussion of products in Chapter 19.

Corollary 20.7.3 Suppose A ∈ PC(M ), and y ∈ Ob(A). Suppose

N,N ′ are interval objects in M . Then the pushout morphism

A → A∪y Ξ(Nz|N′x)

obtained by identifying ξ(|0) and y, is a global weak equivalence.

Proof Apply Corollary 20.7.2 with B = Seg(Ξ(Nz |N ′x)).


Corollary 20.7.4 In the situation described above the preceding corol-

lary, the morphism

A → A∪∐

z∈Ztiξ(|0)∐

z∈Z

Ξ(Nz|N′z)

is a global weak equivalence. Given that A → B was a global weak equiv-

alence, the functor

T : A∪∐

z∈Ztiξ(|0)∐

z∈Z

Ξ(Nz|N′z) → Seg(B)


Proof Choose a well-ordering of Z, giving an exhaustion of Z by subsets

Zi indexed by an ordinal i ∈ β. Let Bi ⊂ B be the full subobject whose

object set is f(Ob(A))∪Zi. By transfinite induction we obtain that the

functors

Ti : A ∪∐

z∈Zitiξ(|0)

∐

z∈Zi

Ξ(Nz |N′z) → Seg(Bi)

are global weak equivalences, using the previous corollary at each step.

At the end of the induction we obtain the required statement.

We are now ready to show the preservation of global trivial cofibra-

tions under pushouts.

Theorem 20.7.5 Suppose A → B is an injective trivial cofibration.

Suppose A → C is any morphism in PC(M ). Then the pushout mor-

phism

C → C ∪A B


Proof We first show this statement assuming that all three objects A,

B and C satisfy the Segal conditions. Noting that B → Seg(B) is an

isomorphism on sets of objects and applying Lemma 16.3.4, it suffices

to show that the map

C → C ∪A Seg(B)

is a global weak equivalence. Define

F := A ∪∐

z∈Ztiξ(|0)∐

z∈Z

Ξ(Nz |N′z),

and consider the map T : F → B defined above. By Corollary 20.7.4, T

438 Intervals


By Lemma 16.3.4 it follows that the map

C ∪A F → C ∪A Seg(B)

is a global weak equivalence, so by 3 for 2 it suffices to show that

C → C ∪A F

is a global weak equivalence. But the map A → F is obtained as a trans-

finite composition of pushouts along things of the form ξ(|0) → Ξ(Nz|N′z),

and by Corollary 20.7.3 these pushouts are global weak equivalences.

Thus, the map C → C ∪A F is a global weak equivalence, which finishes

this part of the proof.

Starting with C ← A → B in general, let A′ := Seg(A), then

B′ := Seg(A′ ∪A B), C′ := Seg(A′ ∪A C).

We get a diagram

C ← A → B

C′↓

← A′

↓

→ B′

↓

such that the bottom row satisfies the hypothesis for the first part of

the proof (all objects satisfy the Segal condition and the second map is

a global trivial cofibration), and such that the vertical arrows are global

weak equivalences inducing isomorphisms on sets of objecs. By Lemma

16.3.4, the bottom map in the diagram

C → C′

C ∪A B

↓

→ C′ ∪A′

B′

↓

is a global weak equivalence. The top vertical map is a global weak

equivalence by construction of C′ and the right vertical map is one too,

by the first part of the proof above. By 3 for 2 we conclude that the left

vertical map is a global weak equivalence, as required.

20.8 A versality property 439

20.8 A versality property

The versality properties for the intervals constructed above, yield a sim-

ilar versality property for any cofibrant replacement of E if the target A

is fibrant in the diagram structure Funcc(∆iOb(A)/Ob(A),M ).

Theorem 20.8.1 Suppose A ∈ PC(M ), and suppose A is fibrant as

an object of PCc(Ob(A),M ) where c indicates either the projective, the

Reedy or the injective structures. Let B → E be a cofibrant replacement

in PCc([1],M ), so Ob(B) is still [1] = υ0, υ1. Then if x, y ∈ Ob(A)

are two objects, they project to equivalent objects in τ≤1(A) if and only

if there exists a morphism B → A sending υ0 to x and υ1 to y.

Proof Since τ≤1B = E is the category with two isomorphic objects,

existence of a map B → A sending υ0 to x and υ1 to y implies that x

and y are internally equivalent in A.

Suppose x and y are internally equivalent. If A is a fibrant object for

the Reedy or injective model structures relative to Ob(A), there is a

morphism Ξ(N |N) → A given by Theorem 20.3.1. For the projective

structure use Ξproj(N |N ′) given by Remark 20.3.2 instead. Denote either

of these maps by C → A. Let C ⊂ Seg(C) be the full subcategory

consisting of only the two main objects, but identify Ob(C) with the two

element set [1] = υ0, υ1 = Ob(E). The map C → Seg(C) is a isotrivial

cofibration so

A → A∪C Seg(C)

is an isotrivial cofibration by Theorem 16.3.3, it follows that our map

extends to Seg(C) → A. This now restricts to a map

C → A

sending υ0 to x and υ1 to y. By contractibility, Theorem 20.6.1,

C → E

is a weak equivalence inducing an isomorphism on sets of objects. Choose

a factorization

Ci→ C′

p→ E

where i is a trivial cofibration and p is a trivial fibration in PC([1],M ).

Again our map extends to C′ → A, but now since B is cofibrant and p

is a trivial fibration there is a lifting B → C′ inducing the identity on

the set of objects. We get the required map B → A.

440 Intervals

The importance of this versality property is that it allows us to replace

a global weak equivalence by one which is surjective on sets of objects.

Corollary 20.8.2 Let B → E be a cofibrant replacement in PCc([1],M )

Suppose f : A → C is a global weak equivalence, and suppose C is a fi-

brant object in PCc(Ob(C),M ). Then there exists a pushout A → A′

by a collection of copies of υ0 → B, and a map A′ → C which is a

global weak equivalence and a surjection on sets of objects.

Proof For each object y ∈ Ob(C) choose x ∈ Ob(A) such that f(x)

is internally equivalent to y. For each such pair we get a map B → C

sending υ0 to f(x) and υ1 to y; attaching a copy of B to A by sending υ0to x and then doing this for all objects y we obtain the required pushout

A′ and extension of the map.

21

The model category of M -enrichedprecategories

In this chapter, we finish the proof that the category PC(M ) of M -

enriched precategories, with variable set of objects, has natural injective

and projective model structures. Given the product theorem of Chapter

19 and the discussion of intervals in Chapter 20, the proof presents no

further obstacles. We also show that the Reedy structure PCReedy(M )

is again tractable, left proper and cartesian, allowing us to iterate the

operation.

21.1 A standard factorization

Follow up on Corollary 20.8.2 of the previous chapter, by analyzing

further the case of maps which are surjective on the set of objects.

Lemma 21.1.1 Suppose f : A → B is a morphism in PC(M ) such

that Ob(f) is surjective. Let Ob(f)∗(B) ∈ PC(Ob(A);M ) be the pre-

category obtained by pulling back along Ob(f) : Ob(A) → Ob(B). Then

f factors as

A → Ob(f)∗(B) → B,

where the first map is an isomorphism on sets of objects, and the second

map Ob(f)∗(B) → B satisfies the right lifting property with respect to

any morphism g : U → V such that Ob(g) is injective.


442 The model category of M -enriched precategories

Proof Given a diagram

U → Ob(f)∗(B)

V↓

→ B↓

in order to get a lifting it suffices to choose a lifting on the level of objects

Ob(V) → Ob(Ob(f)∗(B)) = Ob(A).

This is possible since Ob(q) is injective and Ob(f) surjective.

Corollary 21.1.2 In the situation of the lemma, if I is a set of mor-

phisms in PC(M ) which are all injective on the level of objects, and if

the first map A → Ob(f)∗(B) is in inj(I) then f ∈ inj(I).

Proof Combine the lifting property for inj(I) with the one of the pre-

vious lemma.

21.2 The model structures

We will be applying Theorem 9.9.7 of Chapter 9 to construct the model

structure on PC(M ).

We fix a class of cofibrations denoted generically by c, with c = proj

or c = Reedy. This choice determines the corresponding notions of cofi-

brations in PCc(M ) or PCc(X ;M ). Let I be a set of generators for the

c-cofibrations in PCc(M ), as discussed in Chapter 15. We can choose I

to consist of maps with c-cofibrant domains, for the Reedy and projective

structures, see Chapter 15.

Let Kloc denote a set of morphisms which are pseudo-generators for

the local weak equivalences in PCc([k],M ), as from Theorem 14.1.1

or Theorem 14.3.2. We may assume that they are c-cofibrations, with

c-cofibrant domains if c is Reedy or projective.

Recall that E = Υ(∗) is the category with a single non-identity mor-

phism υ0 → υ1, and E is the category obtained by inverting this map,

that is with a single isomorphism υ0 ∼= υ1. Consider the morphism

υ0 → E . Choose a c-cofibrant replacement

υ0, υ1 → P → E

21.2 The model structures 443

in the model category PCc([1],M ), and let

υ0i0→ P

denote the inclusion morphism of a single object. This is still a c-

cofibration inPCc(M ) (because of Condition (AST) in Definition 10.0.9).

Let Kglob := Kloc ∪ i0. Note that the domain of i0 is the single

object precategory υ0 which is c-cofibrant for any of the c.

Theorem 21.2.1 The class of global weak equivalences is pseudo-

generated by Kglob in the sense of construction (PG) of Section 9.9.

Furthermore, I and J satisfy axioms (PGM1)–(PGM6), so they define

tractable left proper model structures by Theorem 9.9.7. For these model

structures, the weak equivalences are the global weak equivalences; the

cofibrations are the projective (resp. injective resp. Reedy) cofibrations,

and the fibrations are the projective (resp. injective resp. Reedy) global

fibrations.

Proof We have to show that Kglob leads to the class of global weak

equivalences via prescription (PG). This amounts to showing that a map

f : X → Y is a global weak equivalence if and only if there exists a

diagram

X → A

Y↓

→ B↓

with the horizontal maps in cell(Kglob) and the right vertical map in

inj(I).

The maps in Kglob are trivial cofibrations in the projective struc-

ture, and the global trivial cofibrations are preserved by pushout (The-

orem 20.7.5) and transfinite composition (Lemma 16.3.5), so the maps

in cell(Kglob) are global trivial cofibrations. By 3 for 2 for global trivial

cofibrations (Proposition 14.6.4) it follows that if there exists a square

diagram as above then f is a global weak equivalence.

Suppose f is a global weak equivalence, we would like to construct a

square as above. Let r : Y → B be the map given by applying the small

object argument to Y with respect to Kloc. Thus B is Kloc-injective. It

follows that it satisfies the Segal conditions, and is a fibrant object in

PCc(Ob(B),M ) (see Theorems 14.1.1 and 14.3.2).

By Corollary 20.8.2 of the preceding Chapter 20, there exists a pushout


X → X ′ by a collection of copies of the map i0 : υ0 → P and an

extension of rf to a global weak equivalence g : X ′ → B which is

surjective on the set of objects. Note that X → X ′ is in cell(Kglob).

Consider the factorization

X ′ → Ob(g)∗B → B

of Lemma 21.1.1 above. The first map is an isomorphism on the set of

objects, so it can be considered as a map in PC(Ob(X ′),M ). Apply

the small object argument for the set Kloc, to the first map to yield a

factorization

X ′ → A → Ob(g)∗B

such that X ′ → A is in cell(Kloc) and A → Ob(g)∗B is a fibration

in PC(Ob(X ′),M ). However, the composed map X ′ → Ob(g)∗B is a

local weak equivalence, so by 3 for 2 in the local model structure, the

map A → Ob(g)∗B is a trivial fibration; hence it is in inj(I). Apply

now Corollary 21.1.2. Note that the factorization of Lemma 21.1.1 for

the map A → B is just

A → Ob(g)∗B → B

where the first map is the same as previously; thus the first map is in

inj(I) and by Corollary 21.1.2 the full map A → B is in inj(I). The

composition

X → X ′ → A

of two maps in cell(Kglob) is again in cell(Kglob). This completes the

verification that our global weak equivalence f satisfies the condition

(PG).

We now verify axioms (PGM1)–(PGM6) needed to apply Theorem

9.9.7 of Chapter 9.

(PGM1)—by hypothesis M is locally presentable, and I and Kglob are

chosen to be small sets of morphisms;

(PGM2)—we have chosen the domains of arrows in I and Kglob to be

cofibrant, and Kglob consists of c-cofibrations in other words it is con-

tained in cof(I);

(PGM3)—the class of global weak equivalences is closed under retracts

by Proposition 14.6.4 of Chapter 14;

(PGM4)—the class of global weak equivalences satisfies 3 for 2 again by

Proposition 14.6.4;

(PGM5)—the class of global trivial c-cofibrations is closed under pushouts

21.3 The cartesian property 445

by Theorem 20.7.5;

(PGM6)—the class of global trivial c-cofibrations is closed under trans-

finite composition, indeed the cofibrations are closed unter transfinite

composition since they have generating sets, see Chapter 15, and the

weak equivalences are too by Lemma 16.3.5.

Theorem 9.9.7 now applies to show that PC(M ) with the given

classes of c-cofibrations, global weak equivalences, hence global trivial

c-cofibrations whence global c-fibrations, is a tractable left proper closed

model category.

21.3 The cartesian property

Lemma 21.3.1 Suppose A → B and C → D are global Reedy cofibra-

tions, with the first one being a weak equivalence. Then the map

A×D ∪A×C B × C → B ×D

is a global trivial Reedy cofibration.

Proof It is a Reedy cofibration by Corollary 15.6.13. We just have to

show that it is a global weak equivalence. By Corollary 19.2.6, the ver-

tical maps in the diagram

A× C → A×D

B × C↓

→ B ×D↓

are global weak equivalences. Applying Theorem 20.7.5 to pushout along

the left vertical global trivial cofibration, then using 3 for 2, it follows

that the map

A×D → A×D ∪A×C B × C

is a global weak equivalence. Then by 3 for 2 using the right vertical

global weak equivalence, the map in the statement of the lemma is a

global weak equivalence.


category. Then the model category PCReedy(M ) of M -enriched precat-

egories with Reedy cofibrations is again a tractable left proper cartesian

model category.


Proof Observe first of all that direct product commutes with colimits

in PC(M ), as can be seen from the explicit description of products and

colimits and using the corresponding condition for M .

Next, note that the map ∅ → ∗ is a Reedy cofibration, from the

definition.

Proposition 15.6.12 gives the cofibrant property of the pushout-product

map; and the previous Lemma 21.3.1 gives the trivial cofibration prop-

erty. This shows that PCReedy(M ) is cartesian.

To finish the proof we need to note that it is tractable. This can be

seen by inspection of the generating cofibrations for the Reedy structure,

given in Proposition 15.6.11.

Of course, the projective model structure is definitely not cartesian.

On the other hand, one can hope to treat the injective model structure.

There is already a problem with tractability: Lurie and Barwick don’t

mention if their constructions of injective model categories preserve the

tractability condition, at least until a most recent version of a paper

in which Barwick states this property. That would clearly be an im-

portant result, giving tractability for PCinj(M ) in general. In the case

of presheaf categories with monomorphisms as cofibrations, of course

this condition becomes automatic. Similarly, it doesn’t seem immedi-

ately clear whether PCinj(M ) will satisfy condition (PROD) in general,

although again this is relatively easy to see for presheaf categories with

monomorphisms as cofibrations.

21.4 Properties of fibrant objects

Julie Bergner made the very interesting observation [36] that one could

give an explicit characterization for the fibrant objects in the case of

Segal categories M = K . We get the same kind of property in general.

Proposition 21.4.1 Let c = proj or c = Reedy. In the model category

PCc(M ) constructed above, an object A with Ob(A) = X is fibrant

if and only if it is fibrant when considered as an object of the model

category PCc(X,M ) of Theorem 14.1.1 or 14.3.2. In turn this condition

is equivalent to saying that A satisfies the Segal conditions, and is fibrant

as an object of the unital diagram model category Funcc(∆oX/X,M ).

Proof Left to the reader in the current version.

For c = proj then, an M -precategory A is fibrant if and only if it

21.5 The model category of strict M -enriched categories 447

satisfies the Segal conditions, and is levelwise fibrant. For c = Reedy the

fibrancy condition is also pretty easy to check: it just means that the

standart matching maps are fibrations in M .

21.5 The model category of strict M -enriched

categories

Dwyer and Kan proposed, in a series of papers, a model category struc-

ture on the category of strict simplicial categories. Their program was

finished by Bergner [33]. Lurie then generalized this to construct a model

category of strict M -enriched categories in [153, Appendix], then used

that to construct the model category of weakly M -enriched precategories

as we have done above.


category. Let Cat(M ) denote the category of strict M -enriched cate-

gories. Define the notion of weak equivalence in the usual way (see Sec-

tion 14.5). Then Cat(M ) has a tractable left proper model structure in

which the generating cofibrations are obtained by free additions of gener-

ating cofibrations of M in the morphism space between any two objects.

There is a Quillen adjunction

Cat(M ) ←→ PCproj(M )

and indeed the model structure on Cat(M ) can be used to generate the

model structure on PCproj(M ). However, Cat(M ) is not in general

cartesian. It follows that any object of PCproj(M ) is equivalent to a

strict M -enriched category.

Proof See Bergner [33] for M = K and Lurie [153] for arbitrary M .

The strictification result, for the case of Tamsamani n-groupoids, was

proven by Paoli [170] using techniques of Catn-groups.

This theorem offers an alternative route to the construction of the

model structure on PCproj(M ), whose proof is somewhat different from

ours. The advantage is that it also gives the model structure on Cat(M )

and hence the strictificaton result; the disadvantage is that it doesn’t give

the cartesian property. The cartesian question has been treated by Rezk

[179] for the case of iterated Rezk categories. We leave it to the reader

to explore these different points of view.

22

Iterated higher categories

The conclusion of Theorem 21.3.2 matches the hypotheses we imposed

that M be tractable, left proper and cartesian. Therefore, we can it-

erate the construction to obtain various versions of model categories

for n-categories and similar objects. This process is inherent in the def-

initions of Tamsamani [206] and Pelissier [171]. Rezk considered the

corresponding iteration of his definition in [179] following Barwick, and

Trimble’s definition is also iterative. Such an iteration is also related to

Dunn’s iteration of the Segal delooping machine [85], and goes back to

the well-known iterative presentation of the notion of strict n-category,

see Bourn [46] for example.

In what follows unless otherwise indicated, the model categoryPC(M )

will mean by definition the Reedy structure PCReedy(M ).

For any n ≥ 0 define by induction PC0(M ) := M and for n ≥ 1

PCn(M ) := PC(PCn−1(M )).

This is the model category of M -enriched n-precategories. Notations for

objects therein will be discussed below.

In the iterated situation, we can introduce the following definition.

Definition 22.0.2 An M -enriched n-precategory A ∈ PCn(M ) sat-

isfies the full Segal condition if it satisfies the Segal condition as an

PCn−1(M )-precategory, and furthermore inductively for any sequence

of objects x0, . . . , xm ∈ Ob(A) the M -enriched (n− 1)-precategory

A(x0, . . . , xm) ∈ PCn−1(M )

satisfies the full Segal condition.

Lemma 22.0.3 If A is a fibrant object in the (iterated Reedy) model

structure on PCn(M ), then A satisfies the full Segal condition.


22.1 Initialization 449

Proof The Segal condition for the PCn−1(M )-precategory comes from

Proposition 21.4.1 (see Theorem 14.3.2). However, if A is fibrant then

the A(x0, . . . , xm) are fibrant in PCn−1(M ) so by induction they also

satisfy the full Segal condition.

22.1 Initialization

Here are a few possible choices for M to start with.

If M = Set is the model category of sets, with cofibrations and

fibrations being arbitrary morphisms and weak equivalences being iso-

morphisms, then PCn(Set) is the model category of n-precats which was

considered in [193] and for which we have now fixed up the proof.

Let ∗ denote the model category with a single object and a single

morphism. Then PC(∗) is Quillen equivalent (by a product-preserving

map) to the model category ∅, ∗ consisting of the emptyset and the

one-element set, where weak equivalences are isomorphisms. Iterating

again,PC2(∗) is Quillen-equivalent toPC(∅, ∗). These are both model

categories of graphs, the first allowing multiple edges between nodes and

the second allowing only zero or one edges between two nodes. The weak

equivalences are defined by requiring isomorphism on the level of π0 i.e.

the set of connected components of a graph. These model categories of

graphs are Quillen-equivalent to Set but have the advantage that the

cofibrations are monomorphisms. They are related to the notion of setoid

in constructive type theory.

In particular, PCn+2(∗) is Quillen-equivalent to PC(Set) and should

perhaps be thought of as the “true” model category of n-categories.

If we start with M = K the Kan-Quillen model category of simpli-

cial sets, then PCn(K ) is the model category of Segal n-precategories

introduced in [117].

One can imagine further constructions starting with M as a category

of diagrams or other such things. Starting with Z/2-equivariant sets

should be useful for considering n-categories with duals.

22.2 Notations

By Lemma 12.7.4, once we start iterating, the hypothesis (DISJ) of

12.7 will be in vigour. Furthermore, most of our examples of starting

categories (even M = ∗) satisfy (DISJ), see Lemma 12.7.5. Whenever

450 Iterated higher categories

such is the case, it is reasonable to introduce an iteration of the notation

An/ of Section 12.7.

One can note, on the other hand, that even based on this notation as

the general framework, most of what was done in [193] and [194] really

used the notation A(x0, . . . , xn) at the crucial places. So, in a certain

sense, the notations we introduce here are not really the fundamental

objects, nonetheless it is convenient to have them for comparison.

In PCn(M ) for any k ≤ m and any multi-index m1, . . . ,mk we

can introduce the notation Am1,...,mk/ ∈ PCn−k(M ) defined by in-

duction on k. At the initial k = 1 (whenever n ≥ 1), by noting that

A ∈ PC(PCn−1(M )) we can use the notation

Am/ ∈ PCn−1(M )

considered in Section 12.7. Then for k ≥ 2 define inductively

Am1,...,mk/ := (Am1,...,mk−1/)mk/ ∈ PCn−k(M ).

For k < n, define on the other hand

Am1,...,mk:= Ob(Am1,...,mk/) ∈ Set.

One can remark that the notations Am1,...,mk/ ∈ PCn−k(M ) and

Am1,...,mk∈ Set make sense for k < n because we have seen that

PC(M ) satisfies Condition (DISJ), even if M itself does not.

In a related direction, notice that if M is a presheaf category then

PC(M ) is also a presheaf category, by the discussion of Section 12.7. In-

ductively the same is true ofPCn(M ). If M = Presh(Φ) thenPCn(M ) =

Presh(Cn(Φ)) in the notations of Proposition 12.7.6.

22.3 The case of n-nerves

Start with M = Set with the trivial model structure (as Lurie calls

it [153]), where the weak equivalences are isomorphisms and the cofi-

brations and fibrations are arbitrary maps. Iterating the construction of

Theorem 21.3.2 we obtain the iterated Reedy model category structure

PCn(Set).

The underlying category is the category of presheaves of sets on an

iterated version of the construction of Section 12.7,

PCn(Set) = Presh(Cn(∗)).

22.3 The case of n-nerves 451

The underlying category Cn(∗) may be seen as a quotient of ∆n =

∆ × · · · × ∆, indeed it is the same as the category which was denoted

Θn in [193] and [194]. If A ∈ PCn(Set) then the notation discussed

in the previous section applies, and for any multi-index (m1, . . . ,mk)

with k ≤ n we get a set Am1,...,mk∈ Set. For k = n, the notation

Am1,...,mn := Am1,...,mn/ may be used since the model category M is

equal to Set.

That yields a system of notations coinciding with that of [193], [194],

[196] etc. A slight difference is that for A ∈ PCn(Set), and for any

sequence of objects x0, . . . , xm, what we would be denoting here by

A(x0, . . . , xm) ∈ PCn−1(Set)

was denoted in those preprints by Am/(x0, . . . , xm). We have dropped

the subscript ( )m/ for brevity.

The iterated injective and Reedy model structures coincide in the

case of PCn(Set), by applying Proposition 15.7.2 inductively. A map

A → B is a cofibration if and only if, for any multiindex m1, . . . ,mk

with k < n the map Am1,...,mk→ Bm1,...,mk

is an injection of sets. The

monomorphism condition is not imposed at multiindices of length k = n,

indeed at the top level, all maps of Set are cofibrations for its trivial

model structure. The reader may refer to [193] for a fuller discussion of

the notion of cofibrations.

We call PCn(Set) the category of n-prenerves1; and the objects sat-

isfying the full Segal condition are the n-nerves of Tamsamani [206]. A

fibrant object of PCn(Set) is an n-nerve, indeed it satisfies the Segal

conditions at the last iteration (corresponding to the first element m1 of

a multiindex), and furthermore the n − 1-prenerves A(x0, . . . , xm) are

themselves fibrant in PCn−1(Set) so by induction they also satisfy the

Segal conditions at all of their levels. Taking the disjoint union over all

sequences x0, . . . , xm yields Am/ which is an n− 1-nerve.

At n = 1, the category of 1-prenerves is the category of simplicial sets,

and the 1-nerves are the simplicial sets which are nerves of a 1-category,

that is to say the category of 1-nerves is equivalent to Cat. The process

A 7→ Seg(A) is the generation of a category by generators and relations

discussed in Section 16.8.

1 The objects of PCn(Set) were also called n-precats in [193] [194]


22.4 Truncation and equivalences

The definition of weak equivalence we have adopted for PC(M ) in gen-

eral is designed for enrichment over a general model category M . In

Tamsamani’s original definition of n-nerves, the notion of equivalence

and the truncation operations τ≤k were defined inductively along the

way. So, in case of PCn(Set) there remains the question of equating

these two definitions of equivalences.

For any tractable left proper cartesian model category M , define the

pretruncation

τp≤n : PCn(M ) → PCn(Set)

as the functor induced by τ≤0 : M → Set. Applied to PCn−k(M ) for

any 0 ≤ k ≤ n, this gives a pretruncation functor

τp≤k : PCn(M ) → PCk(Set).

If M = Set and n = k then it is the identity. Recall that for A ∈

PC(M ) the truncation operation was defined by τ≤1(A) := τp≤1(Seg(A)).

Remark 22.4.1 It doesn’t seem to be true in general that the trun-

cation functor from PC(M ) to Cat used starting in Chapter 12 could

be expressed in terms of the generators and relations operation from 1-

prenerves to 1-nerves as

τ≤1(A) ∼= Seg(τp≤1(A)).

Indeed, the operation Seg(A) might alter things in a way which isn’t

seen on the level of 1-truncation.

One should impose the full Segal condition in order to be able to use

the pretruncation.

Proposition 22.4.2 If A is an M -enriched n-precategory which satis-

fies the full Segal condition, then for any k ≤ n the pretruncation τp≤k(A)

is a k-nerve, and these truncations may be composed leading at k = 1 to

the usual truncation τ≤1. They are compatible with direct products.

Suppose Af→ B is a weak equivalence in PCn(M ), and A and B both

satisfy the full Segal condition. Then for any 0 ≤ k ≤ n, the truncation

τp≤k(f) is an equivalence of k-nerves in the sense of [206].

For M = Set, a morphism Af→ B in PCn(Set) between n-nerves,

is a weak equivalence if and only if it is an equivalence of n-nerves in

the sense of [206].

22.4 Truncation and equivalences 453

Proof For n = 1, see Lemma 14.5.1. We still should show the compat-

ibility with composing truncation operations. If P := PC(M ) then we

have defined the truncation τ≤0 : P → Set using the model structure of

P: τ≤0(A) it is the set of morphisms from ∗ to A in ho(P). On the other

hand, we have defined the truncation denoted also τ≤0 : PC(M ) → Set

as sending A to the set of isomorphism classes of τ≤1(A). To show that

they are the same, note first that both are invariant under weak equiva-

lences so we may assume that A is fibrant. Then Homho(P)(∗,A) is the

set of morphisms from ∗ to A, up to the relation of homotopy. The set

of morphisms is just Ob(A), and the relation of homotopy says that x is

equivalent to y if and only if there exists a map from an interval object to

A sending the endpoints to x and y respectively (see Lemma 20.1.3). If

this condition holds then looking at the image of the interval in τ≤1(A)

we conclude that the points x and y go to the same isomorphism class.

In the other direction, if x and y go to isomorphic objects in τ≤1(A)

then by the versality property Theorem 20.3.1 plus the contractibility

of the intervals in question, Theorem 20.6.1, the corresponding maps

x, y : ∗ → A are homotopic. This shows that the two versions of τ≤0(A)

coincide.

Assume that n ≥ 2 and the proposition is known for PCn−1(M ).

For k = 1, the truncation τp≤1(A) is the operation of Lemma 14.5.1

which corresponds to the right truncation when applied to A satisfy-

ing the Segal conditions. For k ≥ 2, we have the truncation functor

τp≤k−1 : PCn−1(M ) → PCk−1(Set), taking weak equivalences between

objects which satisfy the full Segal condition, to weak equivalences. It

follows that if A ∈ PCn(M ) satisfies the full Segal conditions, then

applying τp≤k−1 levelwise to A considered as a diagram from ∆oOb(A)

to PCn−1(M ), it yields a diagram from ∆oOb(A) to PCk−1(Set) which

again satisifes the Segal conditions, as well as the full Segal conditions

levelwise. But τp≤k−1 applied levelwise is by definition τp≤k. This shows

that τp≤k(A) is a k-nerve. The composition of two truncation operations

is again a truncation:

τp≤r(τp≤k(A)) = τp≤r(A)

whenever r ≤ k. By induction they are compatible with direct products.

Suppose Af→ B is a weak equivalence in PCn(M ), and A and B

both satisfy the full Segal condition. Since they satisfy the regular Segal

condition, f is essentially surjective, meaning that τ≤0(A) → τ≤0(B)

is surjective, and induces equivalences A(x, y) → B(f(x), f(y)) for all


pairs of objects x, y ∈ Ob(A). But A(x, y) and B(f(x), f(y)) also sat-

isfy the full Segal condition, so by the inductive statement known for

PCn−1(M ), f induces equivalences of k − 1-nerves

(τ≤kA)(x, y) = τ≤k−1A(x, y)∼→ τ≤k−1(B(f(x), f(y))) = (τ≤kB)(f(x), f(y)).

This now implies that f induces an equivalence of k-nerves from τ≤kA

to τ≤k(B).

In the case M = Set, the above argument works in the other direction

to show that if f is an equivalence of n-nerves in the sense of [206], it is

essentially surjective and, by applying the inductive statement for n−1,

it is also fully faithful.

22.5 The (n + 1)-category nCAT

The cartesian model category structure on PCn(M ) allows us to de-

fine a structure of M -enriched n + 1-category denoted CAT (n;M ).

In particular, starting with M = Set we obtain the n + 1-category

nCAT = CAT (n;Set) which was originally discussed in Chapter 3.

In Section 10.2, starting with a tractable cartesian model category P

we get the strict P-enriched category Enr(P). Recall that the objects

of Enr(P) are the cofibrant and fibrant objects of P, and if X,Y

are two such objects then the morphism object is given by the internal

Hom, Enr(P)(X,Y ) = HomP(X,Y ). The identity and composition

operations are the obvious ones.

This general discussion now applies to the model categoryP = PCn(M ).

Define

nCAT (M ) := Enr(PCn(M )).

Note that nCAT (M ) is a PCn(M )-enriched category, so (with the pre-

viously mentioned confusion of notation)

nCAT (M ) ∈ PC(PCn(M )) = PCn+1(M ).

As nCAT (M ) is a strict PCn(M )-enriched category, its Segal maps

are isomorphisms. Notice that for any fibrant cofibrant objects A,B ∈

PCn(M ),

nCAT (M )(A,B) = HomPCn(M )(A,B)

is also fibrant. By the Segal isomorphisms and the fact that fibrant and

22.5 The (n+ 1)-category nCAT 455

cofibrant objects are preserved by direct product, it follows that for any

sequence A0, . . . ,Am the object

nCAT (M )(A0, . . . ,Am) ∈ PCn(M )

is fibrant. In particular, nCAT (M ) is a projectively fibrant PCn(M )-

enriched precategory.

One unfortunate consequence of the strictness on the first level is that

nCAT (M ) is not Reedy-fibrant. Therefore it isn’t quite correct to write

“nCAT (M ) ∈ Ob((n + 1)CAT (M ))” since nCAT (M ) is not a fibrant

object of PCn+1(M ). Let nCAT (M )→ nCAT ′(M ) denote its fibrant

(and automatically cofibrant) replacement. Then

nCAT ′(M ) ∈ Ob((n+ 1)CAT (M )).

The difference between nCAT (M ) and nCAT ′(M ) was one of the main

obstacles which needed to be overcome in the treatment of limits [194].

If A,B are cofibrant and fibrant M -enriched n-precategories, then a

morphism f : A → B corresponds to an object of then M -enriched n-

precategoryHom(A,B), or equivalently to a 1-morphism in nCAT (M ).

The morphism f is a weak equivalence in PCn(M ) if and only if it is

an internal equivalence in nCAT (M ) i.e. it projects to an isomorphism

in τ≤1(nCAT (M )), and we have an equivalence of categories

τ≤1(nCAT (M )) ∼= ho(PCn(M )).

This compatibility was formulated by Tamsamani in asking for nCAT

[206], and may be proven using the same arguments as in the previous

section.

The above discussion applies with M = Set to give the n+ 1-nerve

nCAT := nCAT (Set) of n-nerves; and to M = K to give the Segal

n+1-category nSeCAT := nCAT (K ) of Segal n-categories. These were

used in [117] to discuss the notion of higher stacks.

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